Infinite Sums and Products
Close form formulas for Euler-Ramanujan infinite products
Infinite sums generating sequences of integers
a(n) = (1/e) * sum_{j>1} j!/(j-n)!^2 [A002720]
a(n) = (1/e^n)*Sum_{j >= 1} j^n * n^j / (j-1)! [A299824]
a(n) = (1/e^n)*Sum_{j >= 1} j^n * n^j / (j-1)! [A299824]
1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114, 234662231, 3405357682, 53334454417, 896324308634, 16083557845279, 306827170866106, 6199668952527617, ...
a(n) = (1/e^n)*Sum_{j >= 1} j^n * n^j / (j-1)! [A299824]
a(n) = (1/e^n)*Sum_{j >= 1} j^n * n^j / (j-1)! [A299824]
a(n) = (1/e^n)*Sum_{j >= 1} j^n * n^j / (j-1)! [A299824]
2, 22, 309, 5428, 115155, 2869242, 82187658, 2661876168, 96202473183, 3838516103310, 167606767714397, 7949901069639228, 407048805012563038, ...
a(n) = (1/e^n)*Sum_{j >= 1} j^n * n^j / (j-3)!
a(n) = (1/e^n)*Sum_{j >= 1} j^n * n^j / (j-1)! [A299824]
a(n) = e^( Sum_{j >= 1} Sum_{k=1, j} ln(j/k)) [A001142]
4, 216, 7371, 239424, ...
a(n) = e^( Sum_{j >= 1} Sum_{k=1, j} ln(j/k)) [A001142]
a(n) = e^( Sum_{j >= 1} Sum_{k=1, j} ln(j/k)) [A001142]
a(n) = e^( Sum_{j >= 1} Sum_{k=1, j} ln(j/k)) [A001142]
1, 1, 2, 9, 96, 2500, 162000, 26471025, 11014635520, 11759522374656, 32406091200000000, 231627686043080250000, 4311500661703860387840000, ...
a(n) = Sum_{j >= 1} Sum_{k=1, j} C(j,k) [A000295]
a(n) = e^( Sum_{j >= 1} Sum_{k=1, j} ln(j/k)) [A001142]
a(n) = Sum_{j >= 1} Sum_{k=1, j} C(j,k) [A000295]
0, 0, 1, 4, 11, 26, 57, 120, 247, 502, 1013, 2036, 4083, 8178, 16369, 32752, 65519, 131054, 262125, 524268, 1048555, 2097130, 4194281, 8388584, 16777191,...
a(n) = (1/e) sum_{j>1}j!^2/(j-8)!^3
a(n) = e^( Sum_{j >= 1} Sum_{k=1, j} ln(j/k)) [A001142]
a(n) = Sum_{j >= 1} Sum_{k=1, j} C(j,k) [A000295]
5, 87, 2971, 1632121, 12962661, 1395857215, 194634226067, 33990369362241, ...