A187646 -id:A187646 - cp's OEIS frontend

A187656 Convolution of the (signless) central Stirling numbers of the first kind (A187646).

1, 2, 23, 472, 14109, 557138, 27417263, 1617536576, 111304630793, 8752522524930, 774271257457719, 76102169738598232, 8227653697751043061, 970337814111625277394, 123968202132756025685151, 17055359730313188973301568

Offset: 0

Emanuele Munarini, Mar 12 2011

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A187646.

seq(sum(abs(combinat[stirling1](2*k,k))*abs(combinat[stirling1](2*(n-k),n-k)),k=0..n),n=0..12);

Table[Sum[Abs[StirlingS1[2k, k]]Abs[StirlingS1[2n - 2k, n - k]], {k, 0, n}], {n, 0, 15}]

makelist(sum(abs(stirling1(2*k,k))*abs(stirling1(2*n-2*k,n-k)),k,0,n),n,0,12);

a(n) = sum(k=0, n, abs(stirling(2*k, k, 1)*stirling(2*(n-k), n-k, 1))); \\ Michel Marcus, May 28 2017

a(n) = Sum_{k=0..n} s(2*k,k)*s(2*n-2*k,n-k).

a(n) ~ n^n * c^(2*n) * 2^(3*n) / (sqrt(Pi*(c-1)*n) * exp(n) * (2*c-1)^n), where c = -LambertW(-1,-exp(-1/2)/2). - Vaclav Kotesovec, May 21 2014

A187658 Binomial convolution of the (signless) central Stirling numbers of the first kind (A187646).

Original entry on oeis.org

1, 2, 24, 516, 16064, 655840, 33157240, 1999679696, 140128848384, 11189643689088, 1003005057594240, 99725721676986240, 10892178742891589792, 1296379044138734510656, 166999512859041432577280, 23149972436862049305233280

Offset: 0

Emanuele Munarini, Mar 12 2011

Cf. A187646, A384495, A384496.

seq(sum(binomial(n,k)*abs(combinat[stirling1](2*k,k))*abs(combinat[stirling1](2*(n-k),n-k)),k=0..n),n=0..12);

Table[Sum[Binomial[n, k]Abs[StirlingS1[2k, k]]Abs[StirlingS1[2n - 2k, n - k]], {k, 0, n}], {n, 0, 15}]

makelist(sum(binomial(n,k)*abs(stirling1(2*k,k))*abs(stirling1(2*n-2*k,n-k)),k,0,n),n,0,12);

a(n) = Sum_{k=0..n} binomial(n,k)*s(2*k,k)*s(2*n-2*k,n-k).

Limit_{n->oo} (a(n)/n!)^(1/n) = -8*LambertW(-1, -exp(-1/2)/2)^2 / (1 + 2*LambertW(-1, -exp(-1/2)/2)) = 9.821629929136511797503... - Vaclav Kotesovec, May 30 2025

A293609 Triangle read by rows, a refinement of the central Stirling numbers of the first kind A187646, T(n, k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 7, 4, 0, 90, 120, 15, 0, 1701, 3696, 1316, 56, 0, 42525, 129780, 84630, 12180, 210, 0, 1323652, 5233404, 5184894, 1492744, 104049, 792, 0, 49329280, 240240000, 326426100, 151251100, 22840818, 852852, 3003, 0

Offset: 0

Peter Luschny, Oct 15 2017

			Triangle starts:
[0]        1
[1]        1,         0
[2]        7,         4,         0
[3]       90,       120,        15,         0
[4]     1701,      3696,      1316,        56,        0
[5]    42525,    129780,     84630,     12180,      210,      0
[6]  1323652,   5233404,   5184894,   1492744,   104049,    792,    0
[7] 49329280, 240240000, 326426100, 151251100, 22840818, 852852, 3003, 0

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Row sums are A187646. T(n, 0) = A007820(n) the central Stirling numbers of the second kind A048993. T(n, n-1) = A001791(n) for n>=1.

Cf. A293616.

for n in [$0..9] do seq(A293616(n, n, k), k=0..n) od;

A293609Row[n_] := If[n==0, {1}, Join[CoefficientList[x^(-n) (1 - x)^(2n) PolyLog[-2n, n, x] /. Log[1 - x] -> 0, x], {0}]];
Table[A293609Row[n], {n, 0, 7}] // Flatten

T(n, k) = A293616(n, n, k) for k = 0..n. The main diagonal in terms of rows (!) of the array of triangles A293616. T_row(n) is row n of triangle A293616(n,.,.), i.e. T_row(0) = [1] is row 0 of A000007, T_row(1) = [1, 0] is row 1 of A173018, T_row(2) = [7, 4, 0] is row 2 of A062253, and so on.

Let h(n) = x^(-n)*(1 - x)^(2*n)*PolyLog(-2*n, n, x) and p(n) the polynomial given by the expansion of h(n) after replacing log(1 - x) by 0. Then T(n, k) is the k-th coefficient of p(n) for 0 <= k < n.

A187654 Binomial cumulative sums of the (signless) central Stirling numbers of the first kind (A187646).

Original entry on oeis.org

1, 2, 14, 262, 7740, 305536, 15061692, 890220752, 61347750704, 4829414749504, 427559293150976, 42047904926171552, 4547772798257998256, 536504774914535869664, 68557641564333466819744, 9433619169586732241895776

Offset: 0

Emanuele Munarini, Mar 12 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A187646.

seq(sum(binomial(n,k)*abs(combinat[stirling1](2*k,k)),k=0..n),n=0..12);

Table[Sum[Binomial[n, k]Abs[StirlingS1[2k, k]], {k, 0, n}], {n, 0, 15}]

makelist(sum(binomial(n,k)*abs(stirling1(2*k,k)),k,0,n),n,0,12);

a(n) = sum(k=0, n, binomial(n, k)*abs(stirling(2*k, k, 1))); \\ Michel Marcus, Aug 03 2021

a(n) = Sum_{k=0..n} binomial(n,k)*s(2k,k).

a(n) ~ exp((2*c-1)/(8*c^2)) * abs(Stirling1(2*n,n)) ~ 2^(3*n-1) * n^n * exp((2*c-1)/(8*c^2)-n) * c^(2*n) / (sqrt(Pi*n*(c-1)) * (2*c-1)^n), where c = -LambertW(-1,-exp(-1/2)/2) = 1.7564312086261696769827376166... - Vaclav Kotesovec, May 21 2014

A187659 Convolution of the (signless) central Stirling numbers of the first kind (A187646) and the central Stirling numbers of the second kind (A007820).

Original entry on oeis.org

1, 2, 19, 333, 8862, 322885, 15061381, 858280605, 57766424400, 4479377168841, 392785285842806, 38393983653735732, 4136603248470746422, 486806030644218961182, 62109988002922704031388, 8537900524822110186179616

Offset: 0

Emanuele Munarini, Mar 12 2011

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A007820, A187646.

seq(sum(abs(combinat[stirling1](2*k,k))*combinat[stirling2](2*(n-k),n-k),k=0..n),n=0..12);

Table[Sum[Abs[StirlingS1[2k, k]]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 15}]

makelist(sum(abs(stirling1(2*k,k))*stirling2(2*n-2*k,n-k),k,0,n),n,0,12);

a(n) = sum(k=0, n, abs(stirling(2*k, k, 1)*stirling(2*(n-k), n-k, 2))); \\ Michel Marcus, May 28 2017

a(n) = Sum_{k=0..n} s(2*k,k)*S(2*n-2*k,n-k).

a(n) ~ n^n * c^(2*n) * 2^(3*n-1) / (sqrt(Pi*(c-1)*n) * exp(n) * (2*c-1)^n), where c = -LambertW(-1,-exp(-1/2)/2). - Vaclav Kotesovec, May 21 2014

A187650 Alternated cumulative sums of the (signless) central Stirling numbers of the first kind (A187646).

Original entry on oeis.org

1, 0, 11, 214, 6555, 262770, 13076765, 777866388, 53853263165, 4254252038764, 377667803463431, 37222867283396314, 4033161189724173207, 476511397553009371918, 60969023704806106263737, 8398605422371512041566888

Offset: 0

Emanuele Munarini, Mar 12 2011

Cf. A187646, A332928.

seq(sum((-1)^(n-k)*abs(combinat[stirling1](2*k,k)),k=0..n),n=0..12);

Table[Sum[(-1)^(n-k)Abs[StirlingS1[2k, k]], {k, 0, n}], {n, 0, 15}]

makelist(sum((-1)^(n-k)*abs(stirling1(2*k,k)),k,0,n),n,0,12);

a(n) = Sum_{k=0..n} (-1)^(n-k)*s(2*k,k).

a(n) ~ 2^(3*n-1) * c^(2*n) * n^(n - 1/2) / (sqrt(Pi*(c-1)) * (2*c-1)^n * exp(n)), where c = -LambertW(-1,-exp(-1/2)/2) = 1.7564312086261696769827376166... - Vaclav Kotesovec, Jul 05 2021

A187652 Alternated binomial cumulative sums of the (signless) central Stirling numbers of the first kind (A187646).

Original entry on oeis.org

1, 0, 10, 194, 5932, 237624, 11820780, 702992968, 48662470640, 3843811669088, 341207224961856, 33627579102171680, 3643463136559851440, 430456189350273371648, 55075003474909952394848, 7586546772496980353804704

Offset: 0

Emanuele Munarini, Mar 12 2011

Cf. A187646.

seq(sum((-1)^(n-k)*binomial(n,k)*abs(combinat[stirling1](2*k,k)),k=0..n),n=0..12);

Table[Sum[(-1)^(n - k)Binomial[n, k]Abs[StirlingS1[2k, k]], {k, 0, n}], {n, 0, 15}]

makelist(sum((-1)^(n-k)*binomial(n,k)*abs(stirling1(2*k,k)),k,0,n),n,0,12);

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*abs(Stirling1(2*k,k)).

a(n) ~ c * d^n * (n-1)!, where d = 8*w^2/(2*w-1), where w = -LambertW(-1,-exp(-1/2)/2) = 1.7564312086261696769827376166... and c = exp((1-2*w)/(8*w^2)) / (2^(3/2)*Pi*sqrt(w-1)) = exp(-1/d) / (2^(3/2)*Pi*sqrt(w-1)) = 0.11686978539934159049334861225275481804523808136863346883911376048... - Vaclav Kotesovec, Jul 05 2021, updated May 25 2025

A187664 Convolution of the (signless) central Lah numbers (A187535) and the (signless) central Stirling numbers of the first kind (A187646).

Original entry on oeis.org

1, 3, 49, 1483, 67615, 4173203, 326208269, 30880075203, 3430574739759, 437145190334383, 62803806114813801, 10038354053796477099, 1766255133182030548351, 339166069936077378326187, 70571377417819411767223541

Offset: 0

Emanuele Munarini, Mar 12 2011

Cf. A187535, A187646.

L := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(sum(L(k)*abs(combinat[stirling1](2*(n-k),n-k)),k=0..n),n=0..12);

L[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[L[k]Abs[StirlingS1[2n - 2k, n - k]], {k, 0, n}], {n, 0, 14}]

L(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(L(k)*abs(stirling1(2*n-2*k,n-k)),k,0,n),n,0,12);

a(n) = Sum_{k=0..n} Lah(2*k,k)*s(2*n-2*k,n-k).

a(n) ~ 2^(4*n - 1/2) * n^(n - 1/2) / (sqrt(Pi) * exp(n)). - Vaclav Kotesovec, May 30 2025

A187666 Binomial convolution of the (signless) central Lah numbers (A187535) and the (signless) central Stirling numbers of the first kind (A187646).

Original entry on oeis.org

1, 3, 51, 1599, 74545, 4654255, 365549495, 34642467783, 3846064986001, 489429448820811, 70208261310969435, 11205444535728231855, 1969021774778391995761, 377672618542009829524551, 78507169034687468202172591

Offset: 0

Emanuele Munarini, Mar 12 2011

Cf. A187535, A187646.

L := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(sum(binomial(n,k)*L(k)*abs(combinat[stirling1](2*(n-k),n-k)),k=0..n),n=0..12);

L[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[Binomial[n,k]L[k]Abs[StirlingS1[2n - 2k, n - k]], {k, 0, n}], {n, 0, 14}]

L(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(binomial(n,k)*L(k)*abs(stirling1(2*n-2*k,n-k)),k,0,n),n,0,12);

a(n) = Sum_{k=0..n} binomial(n,k) * Lah(2k,k) * Stirling1(2n-2k,n-k).

a(n) ~ c * 2^(4*n + 1/2) * n^(n - 1/2) / (sqrt(Pi) * exp(n)), where c = Sum_{k>=0} abs(Stirling1(2*k,k)) / (k! * 2^(4*k+1)) = 0.550990257867992515027936630097897... - Vaclav Kotesovec, May 30 2025

A192662 Floor-Sqrt transform of (signless) central Stirling numbers of the first kind (A187646).

Original entry on oeis.org

1, 1, 3, 15, 82, 518, 3652, 28123, 233733, 2075597, 19542826, 193908574, 2017519282, 21921326573, 247882099197, 2908534759303, 35322473621014, 443010881207381, 5726889928765906, 76175517383629544, 1040964231177762308, 14594191539892866665

Offset: 0

Emanuele Munarini, Jul 07 2011

nonn

Cf. A187646.

Table[Floor[Sqrt[Abs[StirlingS1[2n,n]]]],{n,0,100}]

makelist(floor(sqrt(abs(stirling1(2*n,n)))),n,0,24);

a(n) = floor(sqrt(|Stirling1(2*n,n)|)).

cp's OEIS Frontend

A187656 Convolution of the (signless) central Stirling numbers of the first kind (A187646).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A187658 Binomial convolution of the (signless) central Stirling numbers of the first kind (A187646).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

A293609 Triangle read by rows, a refinement of the central Stirling numbers of the first kind A187646, T(n, k) for n >= 0 and 0 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A187654 Binomial cumulative sums of the (signless) central Stirling numbers of the first kind (A187646).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A187659 Convolution of the (signless) central Stirling numbers of the first kind (A187646) and the central Stirling numbers of the second kind (A007820).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A187650 Alternated cumulative sums of the (signless) central Stirling numbers of the first kind (A187646).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

A187652 Alternated binomial cumulative sums of the (signless) central Stirling numbers of the first kind (A187646).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

A187664 Convolution of the (signless) central Lah numbers (A187535) and the (signless) central Stirling numbers of the first kind (A187646).

Views