cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A218675 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(4*n)/n! * exp(-n*x*A(n*x)^4).

Original entry on oeis.org

1, 1, 5, 51, 817, 18562, 576687, 24203258, 1375038677, 106708683355, 11435867474152, 1708844338589752, 358640659116617571, 106261016900832212139, 44607231638918264608274, 26598477338494285370797703, 22569718290467849884279856477
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 5*x^2 + 51*x^3 + 817*x^4 + 18562*x^5 + 576687*x^6 +...
where
A(x) = 1 + x*A(x)^4*exp(-x*A(x)^4) + 2^2*x^2*A(2*x)^8/2!*exp(-2*x*A(2*x)^4) + 3^3*x^3*A(3*x)^12/3!*exp(-3*x*A(3*x)^4) + 4^4*x^4*A(4*x)^16/4!*exp(-4*x*A(4*x)^4) + 5^5*x^5*A(5*x)^20/5!*exp(-5*x*A(5*x)^4) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A218673, A218674, A218676.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^4,x,k*x)^k/k!*exp(-k*x*subst(A^4,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A218676 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(5*n)/n! * exp(-n*x*A(n*x)^5).

Original entry on oeis.org

1, 1, 6, 71, 1311, 34146, 1207717, 57298282, 3653975784, 316252925221, 37596625187796, 6206102367103899, 1434418185304457039, 466995106832397752352, 215051811411620578152401, 140491107719613466192347681, 130481943378389095603359529403
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 6*x^2 + 71*x^3 + 1311*x^4 + 34146*x^5 + 1207717*x^6 +...
where
A(x) = 1 + x*A(x)^5*exp(-x*A(x)^5) + 2^2*x^2*A(2*x)^10/2!*exp(-2*x*A(2*x)^5) + 3^3*x^3*A(3*x)^15/3!*exp(-3*x*A(3*x)^5) + 4^4*x^4*A(4*x)^20/4!*exp(-4*x*A(4*x)^5) + 5^5*x^5*A(5*x)^25/5!*exp(-5*x*A(5*x)^5) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A218672, A218673, A218674, A218675.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^5,x,k*x)^k/k!*exp(-k*x*subst(A^5,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A219184 O.g.f. satisfies: A(x) = Sum_{n>=0} n^(2*n) * x^n * A(x)^n / n! * exp(-n^2*x*A(x)).

Original entry on oeis.org

1, 1, 8, 112, 2202, 55641, 1724050, 63550446, 2725133134, 133546286188, 7370574862110, 452601918694564, 30610161317492690, 2260721225822606054, 181023122013996360316, 15619416644091171417138, 1444615406376578862379054, 142565035949775130517868740
Offset: 0

Views

Author

Paul D. Hanna, Nov 13 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n / n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 8*x^2 + 112*x^3 + 2202*x^4 + 55641*x^5 + 1724050*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^4*x^2*A(x)^2/2!*exp(-4*x*A(x)) + 3^6*x^3*A(x)^3/3!*exp(-9*x*A(x)) + 4^8*x^4*A(x)^4/4!*exp(-16*x*A(x)) + 5^10*x^5*A(x)^5/5!*exp(-25*x*A(x)) +...
simplifies to a power series in x with integer coefficients.
O.g.f. A(x) satisfies A(x) = G(x*A(x)) where G(x) = A(x/G(x)) begins:
G(x) = 1 + x + 7*x^2 + 90*x^3 + 1701*x^4 + 42525*x^5 + 1323652*x^6 +...+ Stirling2(2*n,n)*x^n +...
so that A(x) = (1/x)*Series_Reversion(x/G(x)).

Crossrefs

Cf. A007820, A219228, A217900, A218681, A218672.

Programs

  • PARI
    {a(n)=local(A=1);for(i=1,n,A=sum(m=0, n, (m^2*x*A)^m/m!*exp(-m^2*x*A+x*O(x^n))));polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

Formula

O.g.f. satisfies: A(x) = Sum_{n>=0} Stirling2(2*n,n) * x^n * A(x)^n.

A222076 O.g.f.: Sum_{n>=0} n^n*(n+2)^n * exp(-n*(n+2)*x) * x^n / n!.

Original entry on oeis.org

1, 3, 23, 320, 6397, 166467, 5338412, 203578776, 9001795829, 452924585465, 25555585227999, 1598279794889076, 109748572718377660, 8209004345714098500, 664396187060996529528, 57853075421585981420208, 5393119810256349152565573, 535908449308064099732283429, 56548822143306498413322880709
Offset: 0

Views

Author

Paul D. Hanna, Feb 24 2013

Keywords

Examples

			O.g.f.: A(x) = 1 + 3*x + 23*x^2 + 320*x^3 + 6397*x^4 + 166467*x^5 +...
where
A(x) = 1 + 3*x*exp(-3*x) + 8^2*exp(-8*x)*x^2/2! + 15^3*exp(-15*x)*x^3/3! + 24^4*exp(-24*x)*x^4/4! + 35^5*exp(-35*x)*x^5/5! +...
is a power series in x with integer coefficients.

Links

Crossrefs

Cf. A217901, A217900, A222077, A222078, A222079.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n,j] * 2^(n-j) * StirlingS2[n+j,n],{j,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n, m^m*(m+2)^m*x^m*exp(-m*(m+2)*x+x*O(x^n))/m!), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+2)^k*x^k/(1+k*(k+2)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+2)^n)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+2)^k * x^k / (1 + k*(k+2)*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+2)^n.
a(n) ~ n^n * 2^(2*n+1/2) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+1)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

A222077 O.g.f.: Sum_{n>=0} n^n*(n+3)^n * exp(-n*(n+3)*x) * x^n / n!.

Original entry on oeis.org

1, 4, 34, 504, 10572, 285408, 9419440, 367571200, 16562241744, 846509123520, 48401180913824, 3061687935718272, 212316590908782336, 16018267935253721088, 1306322033185206970368, 114519518777575592865792, 10740222055670467832259840, 1073051903942317493993088000
Offset: 0

Views

Author

Paul D. Hanna, Feb 24 2013

Keywords

Examples

			O.g.f.: A(x) = 1 + 4*x + 34*x^2 + 504*x^3 + 10572*x^4 + 285408*x^5 +...
where
A(x) = 1 + 4*x*exp(-4*x) + 10^2*exp(-10*x)*x^2/2! + 18^3*exp(-18*x)*x^3/3! + 28^4*exp(-28*x)*x^4/4! + 40^5*exp(-40*x)*x^5/5! +...
is a power series in x with integer coefficients.

Links

Crossrefs

Cf. A217902, A217900, A222076, A222078, A222079.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n,j] * 3^(n-j) * StirlingS2[n+j,n],{j,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n, m^m*(m+3)^m*x^m*exp(-m*(m+3)*x+x*O(x^n))/m!), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+3)^k*x^k/(1+k*(k+3)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+3)^n)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+3)^k * x^k / (1 + k*(k+3)*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+3)^n.
a(n) ~ n^n * 2^(2*n+1) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+3/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

A222078 O.g.f.: Sum_{n>=0} n^n*(n+4)^n * exp(-n*(n+4)*x) * x^n / n!.

Original entry on oeis.org

1, 5, 47, 742, 16357, 459369, 15651260, 626935936, 28872594389, 1503262704775, 87328047029511, 5600639046765690, 393092088068927860, 29974039720132943036, 2467669218502361588472, 218168186315818183909344, 20617367868151866462395205, 2074120178028300166990286691
Offset: 0

Views

Author

Paul D. Hanna, Feb 24 2013

Keywords

Examples

			O.g.f.: A(x) = 1 + 5*x + 47*x^2 + 742*x^3 + 16357*x^4 + 459369*x^5 +...
where
A(x) = 1 + 5*x*exp(-5*x) + 12^2*exp(-12*x)*x^2/2! + 21^3*exp(-21*x)*x^3/3! + 32^4*exp(-32*x)*x^4/4! + 45^5*exp(-45*x)*x^5/5! +...
is a power series in x with integer coefficients.

Links

Crossrefs

Cf. A217903, A217900, A222076, A222077, A222079.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n,j] * 4^(n-j) * StirlingS2[n+j,n],{j,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n, m^m*(m+4)^m*x^m*exp(-m*(m+4)*x+x*O(x^n))/m!), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+4)^k*x^k/(1+k*(k+4)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+4)^n)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+4)^k * x^k / (1 + k*(k+4)*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+4)^n.
a(n) ~ n^n * 2^(2*n+3/2) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

A222079 O.g.f.: Sum_{n>=0} n^n*(n+5)^n * exp(-n*(n+5)*x) * x^n / n!.

Original entry on oeis.org

1, 6, 62, 1040, 24076, 703800, 24786512, 1020779520, 48130232528, 2557117300640, 151180506557280, 9846055638729216, 700523098562671360, 54066239308284456960, 4499576117943522662400, 401709919258066014720000, 38299206898825369235170560, 3883999501445283274005895680
Offset: 0

Views

Author

Paul D. Hanna, Feb 24 2013

Keywords

Comments

From Vaclav Kotesovec, May 22 2014: (Start)
Generally, for p>=1, a(n) = 1/n!*Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+p)^n = Sum_{j=0..n} binomial(n,j) * p^(n-j) * StirlingS2(n+j,n).
a(n) ~ n^n * 2^(2*n+p/2) / (sqrt(2*Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+p/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599...
(End)

Examples

			O.g.f.: A(x) = 1 + 6*x + 62*x^2 + 1040*x^3 + 24076*x^4 + 703800*x^5 +...
where
A(x) = 1 + 6*x*exp(-6*x) + 14^2*exp(-14*x)*x^2/2! + 24^3*exp(-24*x)*x^3/3! + 36^4*exp(-36*x)*x^4/4! + 50^5*exp(-50*x)*x^5/5! +...
is a power series in x with integer coefficients.

Links

Crossrefs

Cf. A217904, A217900, A222076, A222077, A222078.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n,j] * 5^(n-j) * StirlingS2[n+j,n],{j,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n, m^m*(m+5)^m*x^m*exp(-m*(m+5)*x+x*O(x^n))/m!), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+5)^k*x^k/(1+k*(k+5)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+5)^n)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+5)^k * x^k / (1 + k*(k+5)*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+5)^n.
a(n) ~ n^n * 2^(2*n+2) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+5/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

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

Views

Author

Paul D. Hanna, Feb 24 2013

Keywords

Examples

			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.

Links

Crossrefs

Cf. A220955, A217900, A007820, A242373.

Programs

  • Mathematica
    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 *)
  • PARI
    {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), ", "))
  • PARI
    {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), ", "))
  • PARI
    {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), ", "))

Formula

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

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

Views

Author

Alois P. Heinz, May 30 2015

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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

A213193 O.g.f.: Sum_{n>=0} (4*n+1)^(4*n+1) * exp(-(4*n+1)^4*x) * x^n / n!.

Original entry on oeis.org

1, 3124, 191757120, 49208861869440, 33030777426968816640, 45829974166034718596428800, 114009204539207742166715857223680, 462192193445890293982679086838571270144, 2851153321165202191241172917762717987236478976
Offset: 0

Views

Author

Paul D. Hanna, Mar 01 2013

Keywords

Comments

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)

Examples

			O.g.f.: A(x) = 1 + 3124*x + 191757120*x^2 + 49208861869440*x^3 +...
where
A(x) = exp(-x) + 5^5*x*exp(-5^4*x) + 9^9*exp(-9^4*x)*x^2/2! + 13^13*exp(-13^4*x)*x^3/3! + 17^17*exp(-17^4*x)*x^4/4! + 21^21*exp(-21^4*x)*x^5/5! +...
is a power series in x with integer coefficients.

Crossrefs

Cf. A222525, A220955, A221214, A217900, A007820, A242449.

Programs

  • Mathematica
    Table[1/n!*Sum[(-1)^(n-k)*Binomial[n,k]*(4*k+1)^(4*n+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[Binomial[4*n+1,n+k]*4^(n+k)*StirlingS2[n+k,n],{k,0,3*n+1}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
  • PARI
    {a(n)=polcoeff(sum(k=0, n, (4*k+1)^(4*k+1)*exp(-(4*k+1)^4*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=(1/n!)*polcoeff(sum(k=0, n, (4*k+1)^(4*k+1)*x^k/(1+(4*k+1)^4*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(4*k+1)^(4*n+1))}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} (4*k+1)^(4*k+1) * x^k / (1 + (4*k+1)^4*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (4*k+1)^(4*n+1).
a(n) ~ n^(3*n+1/2) * 2^(16*n+9/2) / (sqrt(2*Pi*(1-r)) * exp(3*n) * r^(n+1/4) * (4-r)^(3*n+1)), where r = -LambertW(-4*exp(-4)) = 0.0793096051271136564391... . - Vaclav Kotesovec, May 13 2014
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