A351135
a(n) = Sum_{k=0..n} k! * k^(k*n) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 31, 117716, 103060088854, 35762522985456876854, 7426384178533125493811949517898, 1294894823429942179301223205449027573956692920, 253092741940931724343266089700550691376738432767085871485096840
Offset: 0
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a[0] = 1; a[n_] := Sum[k! * k^(k*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 9, 0] (* Amiram Eldar, Feb 02 2022 *)
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a(n) = sum(k=0, n, k!*k^(k*n)*stirling(n, k, 1));
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my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, log(1+k^k*x)^k)))
A350722
a(n) = Sum_{k=0..n} k! * k^(k+n) * Stirling2(n,k).
Original entry on oeis.org
1, 1, 33, 4567, 1652493, 1235777551, 1656820330173, 3619858882041487, 12034209740498292093, 57813156798714532953391, 385490564193781368103929213, 3454086424032897924417605526607, 40500898779980258599522326286912893
Offset: 0
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a[0] = 1; a[n_] := Sum[k! * k^(k+n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 13, 0] (* Amiram Eldar, Feb 03 2022 *)
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a(n) = sum(k=0, n, k!*k^(k+n)*stirling(n, k, 2));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(k*x)-1))^k)))
A351180
a(n) = Sum_{k=0..n} k^(k+n) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 15, 635, 53112, 7367444, 1529130770, 443685287576, 171495189203456, 85174828026304824, 52856314387144232184, 40077340463437963801752, 36457068309928364981668848, 39186634107857517367884040632
Offset: 0
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a[0] = 1; a[n_] := Sum[k^(k + n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 04 2022 *)
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a(n) = sum(k=0, n, k^(k+n)*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*log(1+k*x))^k/k!)))
A351182
a(n) = Sum_{k=0..n} k^(2*k) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 15, 683, 61332, 9135004, 2035708760, 634172615600, 263166948202080, 140322186951905736, 93484350581344936344, 76095870609142447018152, 74311960997497053384537408, 85748280952260853814490688656
Offset: 0
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a(n) = sum(k=0, n, k^(2*k)*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^2*log(1+x))^k/k!)))
A351280
a(n) = Sum_{k=0..n} k! * k^k * Stirling1(n,k).
Original entry on oeis.org
1, 1, 7, 140, 5254, 318854, 28455182, 3506576856, 570360248856, 118356589567440, 30512901324706608, 9566812017770347152, 3584662956711860108352, 1581905384865801328253712, 812047187127758913474118032, 479763784808095613489811245568
Offset: 0
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a[0] = 1; a[n_] := Sum[k! * k^k * StirlingS1[n, k], {k, 1, n}]; Array[a, 16, 0] (* Amiram Eldar, Feb 06 2022 *)
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a(n) = sum(k=0, n, k!*k^k*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*log(1+x))^k)))
A350719
a(n) = Sum_{k=0..n} k! * 2^k * k^n * Stirling1(n,k).
Original entry on oeis.org
1, 2, 30, 1108, 76372, 8463328, 1375868768, 308440047648, 91189383264864, 34376022491122368, 16093445542120281792, 9160424435706947112576, 6230035512106223752576896, 4989402076922846372194268160, 4647526704475074504983564884992
Offset: 0
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a[0] = 1; a[n_] := Sum[k! * 2^k * k^n * StirlingS1[n, k], {k, 1, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 03 2022 *)
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a(n) = sum(k=0, n, k!*2^k*k^n*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*log(1+k*x))^k)))
A350720
a(n) = Sum_{k=0..n} k! * 3^k * k^n * Stirling1(n,k).
Original entry on oeis.org
1, 3, 69, 3948, 422082, 72567522, 18304992558, 6367730357160, 2921446409138136, 1709074810258369776, 1241694104839498851552, 1096850187800368469477424, 1157691464039682741551221152, 1438880771284303822650674399664
Offset: 0
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a[0] = 1; a[n_] := Sum[k! * 3^k * k^n * StirlingS1[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 03 2022 *)
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a(n) = sum(k=0, n, k!*3^k*k^n*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*log(1+k*x))^k)))
A350725
a(n) = Sum_{k=0..n} k! * k^(n-k) * Stirling1(n,k).
Original entry on oeis.org
1, 1, 1, -4, -2, 274, -3442, -12552, 2108664, -63083232, 87416112, 112192496976, -7487840132544, 174521224997040, 19793498724358032, -3109195219736188416, 209306170972547346816, 2973238556525799866496, -3013574861684426837113728, 456220653756733889826621696
Offset: 0
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a[0] = 1; a[n_] := Sum[k! * k^(n-k) * StirlingS1[n, k], {k, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 03 2022 *)
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a(n) = sum(k=0, n, k!*k^(n-k)*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, log(1+k*x)^k/k^k)))
A354674
a(n) = Sum_{k=0..n} k! * k^(k+n) * |Stirling1(n,k)|.
Original entry on oeis.org
1, 1, 33, 4568, 1653010, 1236180194, 1657339714418, 3620923498508952, 12037504737979759944, 57827877567223173191712, 385581993722741959459382352, 3454851578510897594456017095504, 40509304222426523176427339597382336
Offset: 0
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a(n) = sum(k=0, n, k!*k^(k+n)*abs(stirling(n, k, 1)));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k*log(1-k*x))^k)))
Showing 1-9 of 9 results.