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)));
A353254
Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (j - 2 * x).
Original entry on oeis.org
1, 0, -2, -2, 0, 0, -12, -88, -608, -4664, -40032, -381200, -3993520, -45685472, -566975456, -7589393568, -109019255360, -1673050977024, -27321358963904, -473094230383616, -8659054324278528, -167044915214322816, -3387793305708038400, -72061754672510128384
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)));
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)));
A353289
a(n) = Sum_{k=0..floor(n/2)} (n-k)^k * |Stirling1(n-k,k)|.
Original entry on oeis.org
1, 0, 1, 2, 10, 51, 323, 2354, 19535, 181606, 1869549, 21110063, 259400501, 3445913273, 49207968328, 751698726580, 12231484211240, 211208935989003, 3857425360784596, 74292198980174828, 1504832580013205275, 31980327844846620785, 711498612995378484414
Offset: 0
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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+k*x)))
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a(n) = sum(k=0, n\2, (n-k)^k*abs(stirling(n-k, k, 1)));
Showing 1-4 of 4 results.