A226775 -id:A226775 - cp's OEIS frontend

A384472 a(n) = Sum_{k=0..n} binomial(n,k)^3 * Stirling2(2k,k) Stirling2(2n-2k,n-k).

1, 2, 22, 558, 25506, 1770300, 166190354, 19647687682, 2798281247682, 466166725448544, 88942246964278060, 19127775950813311232, 4578817457796314714502, 1207681779462031251096888, 348018457509475159702959174, 108798555057988053563408904750, 36676526343321856806298038370210

Offset: 0

Vaclav Kotesovec, May 30 2025

nonn

In general, for m > 1, Sum_{k=0..n} binomial(n,k)^m * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) ~ 2^((m+1)*n + (m-1)/2) * n^(n-(m+1)/2) / (sqrt(m-1) * Pi^((m+1)/2) * (1-w) * exp(n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775.

Cf. A187655 (m=0), A187657 (m=1), A384471 (m=2), A384470.

Cf. A226775.

Table[Sum[StirlingS2[2*k, k]*StirlingS2[2*n-2*k, n-k]*Binomial[n, k]^3, {k, 0, n}], {n, 0, 20}]

a(n) ~ 2^(4*n + 1/2) * n^(n-2) / (Pi^2 * (1-w) * exp(n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...

A220955 O.g.f.: Sum_{n>=0} (2n+1)^(2n+1) * exp(-(2n+1)^2x) * x^n / n!.

Original entry on oeis.org

1, 26, 1320, 99288, 9901920, 1230768704, 183260197120, 31800433551744, 6301891570411008, 1404224096732154880, 347532097449969496064, 94584986134590717358080, 28076463606243146379018240, 9027122730610037995425792000, 3125219575155651450096795648000

Offset: 0

Paul D. Hanna, Feb 27 2013

nonn

From Vaclav Kotesovec, May 13 2014: (Start)
Generally, for p>1, a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (p*k+1)^(p*n+1) = Sum_{k=0..(p-1)*n+1} p^(n+k) * binomial(p*n+1,n+k) * stirling2(n+k,n).
a(n) ~ n^(n*p-n+1/2) * p^(2*p*n+2+1/p) / (sqrt(2*Pi*(1-r)) * exp((p-1)*n) * r^(n+1/p) * (p-r)^(n*p-n+1)), where r = -LambertW(-p*exp(-p)).
(End)

			O.g.f.: A(x) = 1 + 26*x + 1320*x^2 + 99288*x^3 + 9901920*x^4 +...
where A(x) = exp(-x) + 3^3*exp(-3^2*x)*x + 5^5*exp(-5^2*x)*x^2/2! + 7^7*exp(-7^2*x)*x^3/3! + 9^9*exp(-9^2*x)*x^4/4! + 11^11*exp(-11^2*x)*x^5/5! +...
is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..290

Cf. A222525, A221214, A213193, A217900, A007820, A242449.

Table[1/n! * Sum[(-1)^(n-k)*Binomial[n,k] * (2*k+1)^(2*n+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[Binomial[2*n+1,n+k]*2^(n+k)*StirlingS2[n+k,n],{k,0,n+1}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)

{a(n)=polcoeff(sum(k=0, n, (2*k+1)^(2*k+1)*exp(-(2*k+1)^2*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=(1/n!)*polcoeff(sum(k=0, n, (2*k+1)^(2*k+1)*x^k/(1+(2*k+1)^2*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(2*k+1)^(2*n+1))}
for(n=0, 20, print1(a(n), ", "))

a(n) = 1/n! * [x^n] Sum_{k>=0} (2*k+1)^(2*k+1) * x^k / (1 + (2*k+1)^2*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (2*k+1)^(2*n+1).
a(n) = Sum_{k=0..n+1} 2^(n+k) * binomial(2*n+1,n+k) * stirling2(n+k,n). - Vaclav Kotesovec, May 13 2014
a(n) ~ n^(n+1/2) * 2^(4*n+5/2) / (sqrt(2*Pi*(1-r)) * exp(n) * r^(n+1/2) * (2-r)^(n+1)), where r = -LambertW(-2*exp(-2)) = 0.4063757399599599... (see A226775 = -r) . - Vaclav Kotesovec, May 13 2014

A247238 a(n) = Stirling2(2*n+1, n).

Original entry on oeis.org

1, 15, 301, 7770, 246730, 9321312, 408741333, 20415995028, 1144614626805, 71187132291275, 4864251308951100, 362262620784874680, 29206898819153109600, 2534474684137526739000, 235535731151727520125765, 23339590705557273894321960

Offset: 1

Vladimir Kruchinin, Nov 28 2014

nonn

			O.g.f.: A(x) = x + 15*x^2 + 301*x^3 + 7770*x^4 + 246730*x^5 + 9321312*x^6 + ... where A(x) = 1^3*x*exp(-1^2*x) + 2^5*exp(-2^2*x)*x^2/2! + 3^7*exp(-3^2*x)*x^3/3! + 4^9*exp(-4^2*x)*x^4/4! + 5^11*exp(-5^2*x)*x^5/5! + ...

G. C. Greubel, Table of n, a(n) for n = 1..340

Cf. A048993, A007820, A217899, A243227, A226775.

Table[StirlingS2[2*n+1, n], {n, 1, 20}] (* Vaclav Kotesovec, Nov 29 2014 *)

vector(50, n, stirling(2*n+1, n, 2)) \\ Colin Barker, Nov 28 2014

a(n) = A243227(n) / (n-1)!. - Vaclav Kotesovec, Nov 29 2014
a(n) ~ 2^(2*n+1/2) * n^(n+1/2) / (sqrt(Pi) * sqrt(1-c) * exp(n) * c^n * (2-c)^(n+1)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... (see A226775). - Vaclav Kotesovec, Nov 29 2014
O.g.f. Sum_{n>=1} n^(2*n+1) * x^n * exp(-n^2*x) / n! = Sum_{n>=1} a(n)*x^n. - Paul D. Hanna, Oct 09 2023

A304861 O.g.f. A(x) satisfies: 0 = [x^n] exp( n(n-1) Integral 1/A(x) dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 0, 2, 20, 328, 7664, 231744, 8560512, 372339840, 18593869184, 1046764673152, 65518908623360, 4510397034460160, 338534873778165760, 27505042556295458816, 2404499023598887772160, 225014884122460397678592, 22441327480906466274779136, 2376060993772932821157273600, 266169866452350363506325897216, 31451236460722731478509841711104

Offset: 0

Paul D. Hanna, Jun 01 2018

nonn

Note: 0 = [x^n] exp( (n-1) * Integral 1/F(x) dx ) / F(x) holds for n > 0 when F(x) = sqrt(1 + x^2).
Note: 0 = [x^n] exp( n * Integral 1/G(x) dx ) / G(x) holds for n > 0 when G(x) = 1 + x.
It is remarkable that this sequence should consist entirely of integers.

			O.g.f.: A(x) = 1 + 2*x^2 + 20*x^3 + 328*x^4 + 7664*x^5 + 231744*x^6 + 8560512*x^7 + 372339840*x^8 + 18593869184*x^9 + 1046764673152*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in  exp(n*(n-1) * Integral 1/A(x) dx)/A(x) begins:
n=0: [1, 0, -2, -20, -324, -7584, -230040, -8516976, ...];
n=1: [1, 0, -2, -20, -324, -7584, -230040, -8516976, ...];
n=2: [1, 2, 0, -24, -380, -8424, -248640, -9062720, ...];
n=3: [1, 6, 16, 0, -480, -10528, -292544, -10293696, ...];
n=4: [1, 12, 70, 236, 0, -13472, -378336, -12576960, ...];
n=5: [1, 20, 198, 1260, 5176, 0, -485520, -16616864, ...];
n=6: [1, 30, 448, 4400, 31176, 151792, 0, -21316608, ...];
n=7: [1, 42, 880, 12216, 125340, 989384, 5588416, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp( n*(n-1) * Integral 1/A(x) dx ) / A(x), for n > 0.
RELATED SERIES.
1/A(x) = 1 - 2*x^2 - 20*x^3 - 324*x^4 - 7584*x^5 - 230040*x^6 - 8516976*x^7 - 371005040*x^8 - 18545507840*x^9 - 1044727771680*x^10 + ...
exp(Integral 1/A(x) dx) = 1 + 2*x/2 + 2*x^2/2^2 - 4*x^3/2^3 - 90*x^4/2^4 - 2244*x^5/2^5 - 85196*x^6/2^6 - 4372040*x^7/2^7 - 281105594*x^8/2^8 - 21659046420*x^9/2^9 + ...
exp(2 * Integral 1/A(x) dx) = 1 + 2*x + 2*x^2 - 12*x^4 - 152*x^5 - 2808*x^6 - 71040*x^7 - 2265680*x^8 - 86833824*x^9 - 3878209440*x^10 - 197532405760*x^11 + ..., an integer series.
A'(x)/A(x) = 4*x + 60*x^2 + 1304*x^3 + 38120*x^4 + 1385344*x^5 + 59770928*x^6 + 2973371104*x^7 + 167126930016*x^8 + 10457452841984*x^9 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..520

Cf. A304862, A305144, A305145, A305146, A305147.

Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.

{a(n) = my(A=[1],m); for(i=1,n+1, m=#A; A=concat(A,0); A[m+1] = Vec( exp(m*(m-1)*intformal(1/Ser(A))) / Ser(A) )[m+1] );A[n+1]}
for(n=0,20,print1(a(n),", "))

a(n) ~ sqrt(1-c) * 2^(2*n - 2) * n^(n - 1/2) / (sqrt(Pi) * exp(n) * c^(n - 1/2) * (2-c)^(n-1)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, Oct 18 2020

A222525 O.g.f.: Sum_{n>=0} (2n+1)^(2n) * exp(-(2n+1)^2x) * x^n / n!.

Original entry on oeis.org

1, 8, 232, 12160, 929376, 93590784, 11709432064, 1751777730560, 305065968649728, 60623947402670080, 13538933075023376384, 3356940619048979988480, 915040828127405123420160, 271974910674004076827115520, 87543520972441760055430348800, 30337462571518006406505729884160

Offset: 0

Paul D. Hanna, Feb 24 2013

nonn

			O.g.f.: A(x) = 1 + 8*x + 232*x^2 + 12160*x^3 + 929376*x^4 + 93590784*x^5 +...
where
A(x) = exp(-x) + 3^2*exp(-3^2*x)*x + 5^4*exp(-5^2*x)*x^2/2! + 7^6*exp(-7^2*x)*x^3/3! + 9^8*exp(-9^2*x)*x^4/4! + 11^10*exp(-11^2*x)*x^5/5! +...
is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..290

Cf. A220955, A217900, A007820, A242373.

Table[1/n!*Sum[(-1)^(n-k)*Binomial[n,k]*(2*k+1)^(2*n),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[2^k*Binomial[2*n,k]*StirlingS2[k,n],{k,n,2*n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)

{a(n)=polcoeff(sum(k=0, n, (2*k+1)^(2*k)*exp(-(2*k+1)^2*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=(1/n!)*polcoeff(sum(k=0, n, (2*k+1)^(2*k)*x^k/(1+(2*k+1)^2*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(2*k+1)^(2*n))}
for(n=0, 20, print1(a(n), ", "))

a(n) = 1/n! * [x^n] Sum_{k>=0} (2*k+1)^(2*k) * x^k / (1 + (2*k+1)^2*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (2*k+1)^(2*n).
a(n) = Sum_{k=0..n} 2^(n+k) * binomial(2*n,n+k) * stirling2(n+k,n). - Vaclav Kotesovec, May 13 2014
a(n) ~ 2^(4*n) * n^(n-1/2) / (sqrt(Pi*r*(1-r)) * exp(n) * (r*(2-r))^n), where r = -LambertW(-2*exp(-2)) = 0.4063757399599... (see A226775 = -r). - Vaclav Kotesovec, May 13 2014

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)).

A258467 Number of partitions of 2n into parts of exactly n sorts which are introduced in ascending order.

Original entry on oeis.org

1, 2, 12, 130, 2216, 52078, 1558219, 56524414, 2406802476, 117575627562, 6478447651345, 397345158550386, 26842747368209994, 1980156804133210116, 158365138356099680582, 13647670818304698139989, 1260732993182758276252088, 124273946254095006307105363

Offset: 0

Alois P. Heinz, May 30 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A256130, A187655, A187657, A217899, A217900, A245109, A319732.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(2*n, n):
seq(a(n), n=0..20);

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; a[n_] := T[2n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 06 2017, translated from Maple *)

a(n) = A256130(2n,n).
a(n) ~ 2^(2*n-1/2) * n^(n-1/2) / (sqrt(Pi*(1-c)) * exp(n) * c^n * (2-c)^n), where c = -A226775 = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 31 2015
a(n) ~ Stirling2(2*n, n) = A007820(n). - Vaclav Kotesovec, Jun 01 2015

A288312 Number of endofunctions on [2n] such that the image size equals n.

Original entry on oeis.org

1, 2, 84, 10800, 2857680, 1285956000, 880599202560, 853262368358400, 1111400775560275200, 1873276460474747328000, 3967400888465895264384000, 10313998054713896966296473600, 32291970618091110826769565696000, 119851615755915509174015455948800000

Offset: 0

Alois P. Heinz, Jun 07 2017

nonn

			a(1) = 2: (1,1), (2,2).

Alois P. Heinz, Table of n, a(n) for n = 0..195

Cf. A000142, A048993, A090657, A101817, A219859.

b:= proc(n, k) option remember; `if`(k=n, n!,
      `if`(k=0, 0, n*(b(n-1, k-1)+b(n-1, k)*k/(n-k))))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..15);

Table[StirlingS2[2*n, n]*(2*n)!/n!, {n, 0, 20}] (* Vaclav Kotesovec, Jun 10 2017 *)

a(n)=stirling(2*n, n, 2)*n!*binomial(2*n, n); \\ Indranil Ghosh, Jul 04 2017

from mpmath import *
mp.dps=100
def a(n): return int(stirling2(2*n, n)*fac(n)*binomial(2*n, n))
print([a(n) for n in range(21)]) # Indranil Ghosh, Jul 04 2017

a(n) = Stirling2(2*n,n) * n! * binomial(2*n,n).
a(n) = A090657(2n,n) = A101817(2n,n) = A219859(2n,n).
a(n) ~ n^(2*n - 1/2) * 2^(4*n) / (sqrt(Pi*(1-c)) * c^n * (2-c)^n * exp(2*n)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, Jun 10 2017

A317343 E.g.f. A(x) satisfies: [x^n] exp(n^2*x) / A(x)^n = 0 for n >= 1.

Original entry on oeis.org

1, 1, 3, 43, 1945, 178041, 26792971, 5940440563, 1812303908913, 725818277844145, 368664266359757971, 231291789356545214331, 175516846850044015048393, 158383499766971409675254953, 167568661592262324239839114395, 205391322843896611716680298647491, 288693442696565330249751284373878881, 461220054642000786943064376254671287393

Offset: 0

Paul D. Hanna, Jul 26 2018

nonn

It is remarkable that the logarithm of the e.g.f. A(x) should be an integer series.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 43*x^3/3! + 1945*x^4/4! + 178041*x^5/5! + 26792971*x^6/6! + 5940440563*x^7/7! + 1812303908913*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^2*x) / A(x)^n begins:
n=1: [1, 0, -2, -36, -1764, -167280, -25620600, -5737974480, ...];
n=2: [1, 2, 0, -88, -4160, -371328, -55329536, -12201990400, ...];
n=3: [1, 6, 30, 0, -7812, -698184, -97733304, -20791334880, ...];
n=4: [1, 12, 136, 1296, 0, -1171968, -168658176, -33909447168, ...];
n=5: [1, 20, 390, 7220, 113020, 0, -265712600, -55963975600, ...];
n=6: [1, 30, 888, 25704, 709056, 16600320, 0, -84622337280, ...];
n=7: [1, 42, 1750, 72072, 2909340, 112245672, 3684715944, 0, ...];
n=8: [1, 56, 3120, 172640, 9455488, 508540416, 26199517696, 1150524892160, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + x^2 + 6*x^3 + 74*x^4 + 1400*x^5 + 35676*x^6 + 1140328*x^7 + 43740848*x^8 + 1954336608*x^9 + 99561874080*x^10 + ... + A317344(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A317344, A317345.

{a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^2*x +x*O(x^#A)) / Ser(A)^m )[m+1]/m ); n!*A[n+1]}
for(n=0,20,print1(a(n),", "))

a(n) ~ sqrt(1-c) * 2^(2*n - 3/2) * n^(2*n-1) / (exp(2*n) * c^(n - 1/2) * (2-c)^(n-1)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, Aug 06 2018

A384470 a(n) = n! * Sum_{k=0..n} Stirling2(2k,k) Stirling2(2n-2k,n-k) / binomial(n,k).

Original entry on oeis.org

1, 2, 29, 1108, 82924, 10302768, 1917699552, 499332175200, 173242955039616, 77238974345915520, 43027312823342164800, 29285800226400628915200, 23913110797474508388449280, 23071378298963178620672409600, 25964692904608781751347296204800, 33711625062334209438536728660070400

Offset: 0

Vaclav Kotesovec, May 30 2025

nonn

Cf. A187655, A187657, A384471, A384472.

Cf. A226775.

Table[n! * Sum[StirlingS2[2*k, k] * StirlingS2[2*n-2*k, n-k] / Binomial[n, k], {k, 0, n}], {n, 0, 20}]

a(n) ~ 2^(2*n+1) * n^(2*n) / (sqrt(1-w) * exp(2*n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...

cp's OEIS Frontend

A384472 a(n) = Sum_{k=0..n} binomial(n,k)^3 * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k).

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A220955 O.g.f.: Sum_{n>=0} (2*n+1)^(2*n+1) * exp(-(2*n+1)^2*x) * x^n / n!.

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PARI

PARI

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A247238 a(n) = Stirling2(2*n+1, n).

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A304861 O.g.f. A(x) satisfies: 0 = [x^n] exp( n*(n-1) * Integral 1/A(x) dx ) / A(x), for n > 0.

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A222525 O.g.f.: Sum_{n>=0} (2*n+1)^(2*n) * exp(-(2*n+1)^2*x) * x^n / n!.

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PARI

PARI

PARI

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A238258 Decimal expansion of a constant related to A002465.

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A258467 Number of partitions of 2n into parts of exactly n sorts which are introduced in ascending order.

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A288312 Number of endofunctions on [2n] such that the image size equals n.

A384472 a(n) = Sum_{k=0..n} binomial(n,k)^3 * Stirling2(2k,k) Stirling2(2n-2k,n-k).

A220955 O.g.f.: Sum_{n>=0} (2n+1)^(2n+1) * exp(-(2n+1)^2x) * x^n / n!.

A304861 O.g.f. A(x) satisfies: 0 = [x^n] exp( n(n-1) Integral 1/A(x) dx ) / A(x), for n > 0.

A222525 O.g.f.: Sum_{n>=0} (2n+1)^(2n) * exp(-(2n+1)^2x) * x^n / n!.

A384470 a(n) = n! * Sum_{k=0..n} Stirling2(2k,k) Stirling2(2n-2k,n-k) / binomial(n,k).