A211210
a(n) = Sum_{k=0..n} binomial(n, k)*|S1(n, k)|.
Original entry on oeis.org
1, 1, 3, 16, 115, 1021, 10696, 128472, 1734447, 25937683, 424852351, 7554471156, 144767131444, 2971727661124, 65013102375404, 1509186299410896, 37032678328740751, 957376811266995031, 25999194631060525009, 739741591417352081464, 22000132609456951524051
Offset: 0
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Table[Sum[Binomial[n, k] Abs[StirlingS1[n, k]], {k, 0, n}], {n, 0, 20}]
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a(n) = sum(k=0, n, binomial(n, k)*abs(stirling(n, k, 1))); \\ Michel Marcus, May 10 2021
A176153
Triangle T(n,k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n,j), read by rows.
Original entry on oeis.org
1, 1, 1, 1, -1, -1, 1, -8, -2, -2, 1, -23, 43, 19, 19, 1, -49, 301, -199, -79, -79, 1, -89, 1186, -3314, 796, 76, 76, 1, -146, 3529, -22196, 34644, -2400, 2640, 2640, 1, -223, 8793, -100967, 372863, -362529, 3375, -36945, -36945, 1, -323, 19333, -361691, 2466883, -6010901, 3911515, -33509, 329371, 329371
Offset: 0
Triangle begins as:
1;
1, 1;
1, -1, -1;
1, -8, -2, -2;
1, -23, 43, 19, 19;
1, -49, 301, -199, -79, -79;
1, -89, 1186, -3314, 796, 76, 76;
1, -146, 3529, -22196, 34644, -2400, 2640, 2640;
1, -223, 8793, -100967, 372863, -362529, 3375, -36945, -36945;
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Flat(List([0..10], n-> List([0..n], k-> Sum([0..k], j-> (-1)^j*Stirling1(n,n-j)* Binomial(n,j)) ))); # G. C. Greubel, Nov 26 2019
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[(&+[StirlingFirst(n, n-j)*Binomial(n,j): j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 26 2019
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seq(seq( add(combinat[stirling1](n,n-j)*binomial(n,j), j=0..k), k=0..n), n=0..10); # G. C. Greubel, Nov 26 2019
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T[n_, k_]:= Sum[StirlingS1[n, n-j]*Binomial[n, j], {j,0,k}];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
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T(n,k) = sum(j=0,k, stirling(n,n-j,1)*binomial(n,j)); \\ G. C. Greubel, Nov 26 2019
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[[sum((-1)^j*stirling_number1(n, n-j)*binomial(n,j) for j in (0..k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 26 2019
A047798
a(n) = Sum_{k=0..n} C(n,k)*Stirling2(n,k)^2.
Original entry on oeis.org
1, 1, 3, 31, 443, 9006, 241147, 7956579, 318973867, 15061651528, 824029357046, 51526959899570, 3636995712432667, 287053182699020609, 25126145438688593769, 2421761360666327615911, 255466264644678162575691, 29336098320197429601856772
Offset: 0
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List([0..20], n-> Sum([0..n], k-> Binomial(n,k)*Stirling2(n,k)^2 )); # G. C. Greubel, Aug 07 2019
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[(&+[Binomial(n,k)*StirlingSecond(n,k)^2: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 07 2019
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seq(add(binomial(n,k)*stirling2(n, k)^2, k = 0..n), n = 0..20); # G. C. Greubel, Aug 07 2019
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Table[Sum[Binomial[n, k]*StirlingS2[n, k]^2, {k,0,n}], {n,0,20}] (* G. C. Greubel, Aug 07 2019 *)
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{a(n) = sum(k=0,n, binomial(n,k)*stirling(n,k,2)^2)};
vector(20, n, n--; a(n)) \\ G. C. Greubel, Aug 07 2019
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[sum(binomial(n,k)*stirling_number2(n,k)^2 for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 07 2019
A343841
a(n) = Sum{k=0..n} (-1)^(n-k)*binomial(n, k)*Stirling2(n, k).
Original entry on oeis.org
1, 1, -1, -5, 15, 56, -455, -237, 16947, -64220, -529494, 6833608, -8606015, -459331677, 4335744673, 6800310151, -518075832085, 4315086396640, 19931595013738, -812870258798156, 6648395876520816, 46852711038750520, -1752440325584024944, 15485712825845269456
Offset: 0
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a := n -> add((-1)^(n-k)*binomial(n, k)*Stirling2(n, k), k=0..n):
seq(a(n), n = 0..24);
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a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * StirlingS2[n, k], {k, 0, n}]; Array[a, 24, 0] (* Amiram Eldar, May 07 2021 *)
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a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*stirling(n, k, 2)); \\ Michel Marcus, May 07 2021
A308565
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling1(n,k) * k!.
Original entry on oeis.org
1, 1, 0, -6, -12, 140, 1020, -5208, -117264, -2448, 17756640, 117905040, -3177424800, -56997933408, 523176632160, 25824592321920, 31907065317120, -12118922683971840, -151839867298498560, 5619086944920958464, 172859973799199892480, -1989399401447725854720, -170925579909303883614720
Offset: 0
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Table[Sum[Binomial[n, k] StirlingS1[n, k] k!, {k, 0, n}], {n, 0, 22}]
Table[n! SeriesCoefficient[(1 + Log[1 + x])^n, {x, 0, n}], {n, 0, 22}]
A344053
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*Stirling2(n, k)*k!.
Original entry on oeis.org
1, 1, 0, -9, -40, 125, 3444, 18571, -241872, -5796711, -24387220, 1132278191, 25132445832, 8850583573, -10681029498972, -214099676807085, 1643397436986464, 176719161389104817, 2976468247699317468, -71662294521163070153, -4638920054290748840520, -55645074852328083377619
Offset: 0
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a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * StirlingS2[n, k] * k!, {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, May 10 2021 *)
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a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*stirling(n, k, 2)*k!); \\ Michel Marcus, May 10 2021
Showing 1-6 of 6 results.
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