A342110 -id:A342110 - cp's OEIS frontend

A342111 a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n,k) * Stirling1(n,n-k).

1, 0, 1, 12, 193, 3980, 100805, 3034920, 105994833, 4215106728, 188097696345, 9309515255700, 506149663220641, 29989851619249236, 1923467938147053389, 132771455705186298000, 9814431285244231295265, 773520674985391641371280, 64752473306596841023424945

Offset: 0

Vaclav Kotesovec, Feb 28 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A008275, A014322, A047796, A132393, A342110.
Cf. A048994, A155826.
Cf. A129256, A187235, A246117.

[(&+[(-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 03 2021

Table[(-1)^n*Sum[StirlingS1[n, k]*StirlingS1[n, n-k], {k, 0, n}], {n, 0, 20}]

a(n) = (-1)^n*sum(k=0, n, stirling(n, k, 1)*stirling(n, n-k, 1)); \\ Michel Marcus, Feb 28 2021

[sum( stirling_number1(n, k)*stirling_number1(n, n-k) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Jun 03 2021

a(n) ~ c * d^n * (n-1)!, where
d = A238261 = -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)) = 4.9108149645682558987515348052403521978987052817678471761394112...
c = 1/(4*sqrt(-LambertW(-1, -exp(-1/2)/2)) * sqrt(-1 - LambertW(-1, -exp(-1/2)/2))*Pi) = 0.06903826111269387517867145566264007373042059749428879149076344304196548... - Vaclav Kotesovec, Feb 28 2021, updated May 14 2025
a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^2. - Seiichi Manyama, May 13 2025

A047797 a(n) = Sum_{k=0..n} Stirling2(n,k)^2.

Original entry on oeis.org

1, 1, 2, 11, 87, 952, 13513, 237113, 5016728, 125121009, 3615047527, 119384499720, 4455637803543, 186152008588691, 8636436319397292, 441871067839416319, 24781002306869712365, 1515279889256750470086, 100546673139756241189021

Offset: 0

N. J. A. Sloane

nonn

If S is the lower matrix of Stirling numbers of the second kind, this sequence (without the first term 1) is the diagonal of the matrix S.Transpose[S]. - Sergio Falcon, May 02 2007

G. C. Greubel, Table of n, a(n) for n = 0..320

Cf. A008277, A047796, A342110.

List([0..20], n-> Sum([0..n], k-> Stirling2(n,k)^2 )); # G. C. Greubel, Aug 07 2019

[(&+[StirlingSecond(n,k)^2: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 07 2019

seq(add(Stirling2(n, k)^2, k = 0..n), n = 0..20); # G. C. Greubel, Aug 07 2019

Table[Sum[StirlingS2[n,k]^2,{k,0,n}],{n,0,20}] (* Emanuele Munarini, Jul 01 2011 *)

makelist(sum(stirling2(n,k)^2,k,0,n),n,0,20); /* Emanuele Munarini, Jul 01 2011 */

{a(n) = sum(k=0,n, stirling(n,k,2)^2)};
vector(20, n, n--; a(n)) \\ G. C. Greubel, Aug 07 2019

[sum(stirling_number2(n,k)^2 for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 07 2019

A238258 Decimal expansion of a constant related to A002465.

Original entry on oeis.org

3, 0, 8, 8, 2, 7, 7, 3, 0, 4, 7, 4, 1, 7, 4, 0, 1, 7, 9, 1, 1, 5, 8, 4, 0, 0, 8, 2, 0, 2, 5, 4, 3, 8, 2, 7, 6, 8, 3, 6, 4, 4, 4, 8, 9, 7, 1, 4, 2, 0, 1, 3, 8, 7, 6, 7, 2, 4, 7, 7, 3, 0, 1, 2, 1, 7, 6, 5, 1, 6, 8, 1, 2, 7, 8, 8, 2, 6, 6, 6, 6, 9, 5, 2, 0, 3, 2, 7, 1, 1, 3, 0, 9, 6, 1, 9, 4, 6, 0, 0, 9, 7, 3, 0, 9

Offset: 1

Vaclav Kotesovec, Feb 21 2014

			3.08827730474174017911584...

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.249

Cf. A002465, A238260, A187235, A226775, A342110.

RealDigits[N[-2/LambertW[-2/E^2]/(2+LambertW[-2/E^2]), 105]][[1]]

Equals lim n->infinity (A002465(n)/(n-1)!)^(1/n).
Equals -2 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))).
Equals -2 / (A226775 * (2 + A226775)).

cp's OEIS Frontend

A342111 a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n,k) * Stirling1(n,n-k).

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A047797 a(n) = Sum_{k=0..n} Stirling2(n,k)^2.

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A238258 Decimal expansion of a constant related to A002465.

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