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)));
A353253
Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (j - x).
Original entry on oeis.org
1, 0, -1, -1, -1, -3, -14, -76, -480, -3491, -28792, -265708, -2713753, -30395515, -370509784, -4883351213, -69205187838, -1049436525897, -16956113955333, -290817728309779, -5277059794403117, -101005287980087110, -2033813167589257170, -42977173319758429942
Offset: 0
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a[n_] := Sum[(-1)^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-x)))
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a(n) = sum(k=0, n\2, (-1)^k*abs(stirling(n-k, k, 1)));
A352802
Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (j + 3 * x).
Original entry on oeis.org
1, 0, 3, 3, 15, 45, 198, 972, 5652, 37881, 289548, 2492640, 23906475, 253012653, 2930556024, 36883817127, 501315357690, 7318715960511, 114224260779891, 1897913866979529, 33449523840512127, 623265596538965334, 12241892922194658510, 252793167644378784006
Offset: 0
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a[n_] := Sum[3^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+3*x)))
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a(n) = sum(k=0, n\2, 3^k*abs(stirling(n-k, k, 1)));
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));
Showing 1-4 of 4 results.