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)));
A353255
Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (2 * j + x).
Original entry on oeis.org
1, 0, 1, 2, 9, 54, 429, 4252, 50605, 703388, 11184597, 200247446, 3986363597, 87343744490, 2088739037209, 54134344486296, 1511446306795417, 45227224242345336, 1443916049346447913, 48989635949583331658, 1760229264304229244753, 66770472164443344587550
Offset: 0
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a[n_] := Sum[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, 2^(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)));
Showing 1-3 of 3 results.