A353256
Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (3 * j + x).
Original entry on oeis.org
1, 0, 1, 3, 19, 171, 2044, 30528, 547390, 11457237, 274198402, 7385438214, 221099038597, 7282925988615, 261763288109884, 10194190355448399, 427609812103844122, 19220373155515189149, 921621193002227307943, 46958377673245988620737
Offset: 0
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a[n_] := Sum[3^(n-2*k) * Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}]; Array[a, 20, 0] (* Amiram Eldar, Apr 09 2022 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, 3*j+x)))
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a(n) = sum(k=0, n\2, 3^(n-2*k)*abs(stirling(n-k, k, 1)));
A353257
Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (2 * j - x).
Original entry on oeis.org
1, 0, -1, -2, -7, -42, -341, -3452, -41835, -590452, -9511213, -172182182, -3460540075, -76455710870, -1841772619273, -48043721545240, -1349168210580087, -40581475067022120, -1301688751836211065, -44352720153871514858, -1599833618118922360175
Offset: 0
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a[n_] := Sum[(-1)^k * 2^(n - 2*k) * Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}]; Array[a, 20, 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, 2*j-x)))
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a(n) = sum(k=0, n\2, (-1)^k*2^(n-2*k)*abs(stirling(n-k, k, 1)));
A353258
Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (3 * j - x).
Original entry on oeis.org
1, 0, -1, -3, -17, -153, -1846, -27828, -503000, -10599873, -255143728, -6906078108, -207627211745, -6864486246225, -247526246562328, -9667515778323735, -406560434763167342, -18316445888374834635, -880110629723965618045, -44928348211160605056537
Offset: 0
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a[n_] := Sum[(-1)^k * 3^(n - 2*k) * Abs[StirlingS1[n - k, k]], {k, 0, Floor[n/2]}]; Array[a, 20, 0] (* Amiram Eldar, Apr 09 2022 *)
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, 3*j-x)))
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a(n) = sum(k=0, n\2, (-1)^k*3^(n-2*k)*abs(stirling(n-k, k, 1)));
A353290
a(n) = Sum_{k=0..floor(n/2)} (n-k)^(n-2*k) * |Stirling1(n-k,k)|.
Original entry on oeis.org
1, 0, 1, 2, 19, 393, 15177, 939394, 85063260, 10599342278, 1739073390797, 363404567436467, 94224446795779884, 29683590039199285223, 11167286542016941966714, 4945143125245884296040780, 2546112368234517215955646341, 1508197687055444623135714912377
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, k*j+x)))
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a(n) = sum(k=0, n\2, (n-k)^(n-2*k)*abs(stirling(n-k, k, 1)));
Showing 1-4 of 4 results.