A171367
Antidiagonal sums of triangle of Stirling numbers of 2nd kind A048993.
Original entry on oeis.org
1, 0, 1, 1, 2, 4, 9, 22, 58, 164, 495, 1587, 5379, 19195, 71872, 281571, 1151338, 4902687, 21696505, 99598840, 473466698, 2327173489, 11810472444, 61808852380, 333170844940, 1847741027555, 10532499571707, 61649191750137, 370208647200165, 2278936037262610, 14369780182166215
Offset: 0
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b:= proc(n, m) option remember; `if`(n<=m,
`if`(n=m, 1, 0), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..30); # Alois P. Heinz, May 16 2023
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Table[Sum[StirlingS2[n-k, k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 18 2016 *)
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makelist(sum(stirling2(n-k,k),k,0,n),n,0,60); /* Emanuele Munarini, Jun 01 2012 */
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a(n) = sum(k=0, n, stirling(n-k, k,2)); /* Joerg Arndt, Jan 16 2013 */
A097342
a(n) = Sum_{k=0..floor(n/2)} Stirling2(n-k,k) * 3^k.
Original entry on oeis.org
1, 0, 3, 3, 12, 30, 93, 300, 1038, 3810, 14781, 60375, 258807, 1160949, 5435652, 26502555, 134282406, 705720549, 3840542031, 21608662710, 125523530724, 751831408929, 4637611255422, 29428408797852, 191907975348210
Offset: 0
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Table[Sum[StirlingS2[n-k,k]3^k,{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, May 24 2012 *)
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my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, 3^k*x^(2*k)/prod(j=1, k, 1-j*x))) \\ Seiichi Manyama, Apr 09 2022
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a(n) = sum(k=0, n\2, 3^k*stirling(n-k, k, 2)); \\ Seiichi Manyama, Apr 09 2022
A353252
Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (j + 2 * x).
Original entry on oeis.org
1, 0, 2, 2, 8, 24, 100, 488, 2832, 19096, 147296, 1281392, 12422864, 132870368, 1554525152, 19750621216, 270817685568, 3986140113792, 62686410981696, 1048946532137216, 18608550117641728, 348854564104019072, 6891109834644748032, 143058034748452036352
Offset: 0
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a[n_] := Sum[2^k * Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}]; Array[a, 25, 0] (* Amiram Eldar, Apr 09 2022 *)
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my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, j+2*x)))
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a(n) = sum(k=0, n\2, 2^k*abs(stirling(n-k, k, 1)));
A353260
Expansion of Sum_{k>=0} (-1)^k * x^(2*k)/Product_{j=1..k} (1 - j * x).
Original entry on oeis.org
1, 0, -1, -1, 0, 2, 5, 8, 6, -18, -111, -377, -953, -1567, 964, 23411, 133702, 554185, 1801323, 3910514, -2415952, -92788743, -700128734, -3842587204, -17042883146, -57693979779, -86308109341, 779904767601, 10180307035351, 78523141206142, 481780714913151
Offset: 0
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a[n_] := Sum[(-1)^k * StirlingS2[n - k, k], {k, 0, Floor[n/2]}]; Array[a, 30, 0] (* Amiram Eldar, Apr 09 2022 *)
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my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (-1)^k*x^(2*k)/prod(j=1, k, 1-j*x)))
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a(n) = sum(k=0, n\2, (-1)^k*stirling(n-k, k, 2));
A353261
Expansion of Sum_{k>=0} (-2)^k * x^(2*k)/Product_{j=1..k} (1 - j * x).
Original entry on oeis.org
1, 0, -2, -2, 2, 10, 18, 10, -62, -310, -894, -1590, 642, 21514, 120322, 461130, 1230466, 877194, -16158974, -142301750, -798423166, -3397990646, -9764986878, 2009650762, 294960691330, 2788851766154, 18403159253250, 95083470290634, 350847712602498
Offset: 0
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a[n_] := Sum[(-2)^k * StirlingS2[n - k, k], {k, 0, Floor[n/2]}]; Array[a, 30, 0] (* Amiram Eldar, Apr 09 2022 *)
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my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (-2)^k*x^(2*k)/prod(j=1, k, 1-j*x)))
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a(n) = sum(k=0, n\2, (-2)^k*stirling(n-k, k, 2));
A353287
a(n) = Sum_{k=0..floor(n/2)} k^k * Stirling2(n-k,k).
Original entry on oeis.org
1, 0, 1, 1, 5, 13, 56, 223, 1056, 5243, 28401, 163578, 1003332, 6506149, 44464510, 319066188, 2396942740, 18800878491, 153611297283, 1304600660023, 11495292868763, 104907727533628, 990067627794487, 9648859125705064, 96978616443859923
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, k^k*x^(2*k)/prod(j=1, k, 1-j*x)))
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a(n) = sum(k=0, n\2, k^k*stirling(n-k, k, 2));
Showing 1-6 of 6 results.