A091695 -id:A091695 - cp's OEIS frontend

A293012 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1 - x)^k).

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 31, 73, 1, 1, 1, 9, 55, 241, 501, 1, 1, 1, 11, 85, 529, 2261, 4051, 1, 1, 1, 13, 121, 961, 6121, 24781, 37633, 1, 1, 1, 15, 163, 1561, 13041, 82711, 309835, 394353, 1, 1, 1, 17, 211, 2353, 24101, 207001, 1273567, 4342241, 4596553, 1

Offset: 0

Ilya Gutkovskiy, Sep 28 2017

			E.g.f. of column k: A_k(x) =  1 + x/1! + (2*k + 1)*x^2/2! + (3*k^2 + 9*k + 1)*x^3/3! + (4*k^3 + 36*k^2 + 32*k + 1)*x^4/4! + ...
Square array begins:
  1,   1,    1,    1,     1,     1,  ...
  1,   1,    1,    1,     1,     1,  ...
  1,   3,    5,    7,     9,    11,  ...
  1,  13,   31,   55,    85,   121,  ...
  1,  73,  241,  529,   961,  1561,  ...
  1, 501, 2261, 6121, 13041, 24101,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..4 give A000012, A000262, A082579, A091695, A361283.

Main diagonal gives A293013.

Table[Function[k, n! SeriesCoefficient[Exp[x/(1 - x)^k], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

T(n, k) = n!*sum(j=0, n, binomial(n+(k-1)*j-1, n-j)/j!); \\ Seiichi Manyama, Mar 06 2023

E.g.f. of column k: exp(x/(1 - x)^k).
From Seiichi Manyama, Oct 21 2017: (Start)
Let B(j,k) = (-1)^(j-1)*binomial(-k,j-1) for j>0 and k>=0.
A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0. (End)
A(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*j-1,n-j)/j!. - Seiichi Manyama, Mar 06 2023

A361279 Expansion of e.g.f. exp(x * (1+x)^3).

Original entry on oeis.org

1, 1, 7, 37, 241, 1981, 17551, 171697, 1860097, 21609721, 268697431, 3566446621, 50060084977, 740156116597, 11496472967071, 186824483634601, 3167058238988161, 55882288483846897, 1023891003620741287, 19440027237549627541, 381822392009503555441

Offset: 0

Seiichi Manyama, Mar 06 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..508

Column k=3 of A361277.

Cf. A091695.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1+x)^3)))

a(n) = n!*sum(k=0, n, binomial(3*k, n-k)/k!);

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j*binomial(3, j-1)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..n} binomial(3*k,n-k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * binomial(3,k-1) * a(n-k)/(n-k)!.
D-finite with recurrence a(n) -a(n-1) +6*(-n+1)*a(n-2) -9*(n-1)*(n-2)*a(n-3) -4*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Mar 13 2023
a(n) ~ 2^(n/2 - 1) * n^(3*n/4) / exp(3*n/4 - 3*n^(3/4)/2^(3/2) - 15*n^(1/2)/64 + n^(1/4)/2^(19/2) + 27/1024) * (1 + 724053*sqrt(2)/(2621440*n^(1/4))). - Vaclav Kotesovec, Nov 11 2023

A380515 Expansion of e.g.f. exp(xG(x)^3) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 7, 109, 2689, 91261, 3950191, 208064137, 12917499169, 923765042809, 74780847503191, 6760168138392901, 675023676995501857, 73787463232202560309, 8763902701210982610559, 1123850728979698205132641, 154757223522414820829369281, 22775744033825102490806751217

Offset: 0

Seiichi Manyama, Jan 26 2025

nonn

Cf. A380513, A380514, A380516.
Cf. A080893, A380511.
Cf. A091695, A250917, A380512.
Cf. A002293, A006632, A370057, A382059.

a(n) = if(n==0, 1, 3*n!*sum(k=0, n-1, binomial(3*n+k, k)/((3*n+k)*(n-k-1)!)));

a(n) = 3 * n! * Sum_{k=0..n-1} binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.
a(n) = U(1-n, 2-4*n, 1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
E.g.f.: exp( Series_Reversion( x*(1-x)^3 ) ). - Seiichi Manyama, Mar 15 2025

A361599 Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x).

Original entry on oeis.org

1, 2, 11, 88, 881, 10526, 145867, 2294636, 40302593, 780263866, 16483592171, 376901809472, 9265228770481, 243493769839958, 6808261249400171, 201697053847178836, 6308214318127014017, 207622266953125336946, 7170928402389293540683, 259247888385780787392296

Offset: 0

Seiichi Manyama, Mar 17 2023

nonn

Column k=3 of A361600.

Cf. A091695.

Table[n! * Sum[Binomial[n+2*k,3*k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^3)/(1-x)))

a(n) = n! * Sum_{k=0..n} binomial(n+2*k,3*k)/k! = Sum_{k=0..n} (n+2*k)!/(3*k)! * binomial(n,k).
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = 2*(2*n - 1)*a(n-1) - (n-1)*(6*n - 11)*a(n-2) + (n-2)*(n-1)*(4*n - 9)*a(n-3) - (n-3)^2*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(-1/8) * exp(-1/27 - 3^(-5/4)*n^(1/4)/8 + sqrt(n/3)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n + 1/8) / 2 * (1 + (34237/69120)*3^(1/4)/n^(1/4)). (End)

A367789 E.g.f. satisfies A(x) = exp( x/(1-x)^3 * A(x) ).

Original entry on oeis.org

1, 1, 9, 106, 1697, 35076, 893947, 27165706, 960298593, 38751082552, 1758831242291, 88726543365054, 4926355857050641, 298605321687360676, 19623211558172733435, 1389870724939251455506, 105556814502357807727553, 8557797733469700008170224

Offset: 0

Seiichi Manyama, Nov 30 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A362775, A367790.

Cf. A091695, A360100.

A367789 := proc(n)
    n!*add((k+1)^(k-1) * binomial(n+2*k-1,n-k)/k!,k=0..n) ;
end proc:
seq(A367789(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^3))))

E.g.f.: exp( -LambertW(-x/(1-x)^3) ).

a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+2*k-1,n-k)/k!.

A250917 Expansion of e.g.f. exp( xC(x)^3 ) where C(x) = (1 - sqrt(1-4x))/(2*x) is the g.f. of the Catalan numbers, A000108.

Original entry on oeis.org

1, 1, 7, 73, 1033, 18541, 403831, 10351237, 305355793, 10192132153, 379819484551, 15634219476481, 704566985120857, 34506514429777573, 1825081888365736183, 103685565729559782781, 6297505655719537293601, 407233553972252986277617, 27935786938445348562454663

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

In general, if k>0 and e.g.f. = exp(x*C(x)^k) where C(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers, then a(n) ~ k * 2^(2*n + k - 5/2) * n^(n-1) / exp(n - 2^(k-2)). - Vaclav Kotesovec, Aug 22 2017

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 73*x^3/3! + 1033*x^4/4! + 18541*x^5/5! +...
such that log(A(x)) = x*C(x)^3,
log(A(x)) = x + 3*x^2 + 9*x^3 + 28*x^4 + 90*x^5 + 297*x^6 + 1001*x^7 +...
where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Cf. A080893, A251568.
Cf. A091695, A380512, A380515.
Cf. A000108, A001517.

{a(n)=my(C=1); for(i=1, n, C=1+x*C^2 +x*O(x^n));
n!*polcoef(exp(x*C^3), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0, 1, sum(k=0, n, n!/k! * binomial(2*n+k-1, n-k) * 3*k/(n+2*k) ))}
for(n=0, 20, print1(a(n), ", "))

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(serreverse(x*(1-x))^3/x^2))) \\ Seiichi Manyama, Mar 15 2025

a(n) = Sum_{k=0..n} n!/k! * binomial(2*n+k-1, n-k) * 3*k/(n+2*k)  for n>0 with a(0)=1.
a(n) ~ 3 * 2^(2*n+1/2) * n^(n-1) / exp(n-2). - Vaclav Kotesovec, Aug 22 2017
Conjecture D-finite with recurrence: +2*a(n) +(-11*n+20)*a(n-1) +(n^3+9*n^2-116*n+164)*a(n-2) +(-4*n^4+35*n^3+n^2-317*n+342)*a(n-3) -6*(n-3)*(6*n^3-50*n^2+147*n-176)*a(n-4) +12*(n-5)*(2*n-9)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 25 2020
E.g.f.: exp( (1/x)^2 * Series_Reversion( x*(1-x) )^3 ). - Seiichi Manyama, Mar 15 2025

A335345 Expansion of e.g.f. exp(x^2/(2*(1 - x)^3)).

Original entry on oeis.org

1, 0, 1, 9, 75, 690, 7305, 89145, 1237425, 19221300, 329371245, 6157738125, 124551652995, 2707913238030, 62945320162725, 1557291398788125, 40844991621859425, 1131753403094113800, 33025920511859300025, 1012128709342410284625, 32494107983067177522075

Offset: 0

Ilya Gutkovskiy, Jun 02 2020

nonn

Cf. A001809, A091695, A185369, A335344.

nmax = 20; CoefficientList[Series[Exp[x^2/(2 (1 - x)^3)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (1/4) Sum[Binomial[n - 1, k - 1] k (k - 1) k! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

seq(n)=Vec(serlaplace(exp(x^2/(2*(1 - x)^3) + O(x*x^n)))) \\ Andrew Howroyd, Jun 02 2020

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A001809(k) * a(n-k).
D-finite with recurrence 2*a(n) +8*(-n+1)*a(n-1) +2*(n-1)*(6*n-13)*a(n-2) -(n-1)*(n-2)*(8*n-23)*a(n-3) +2*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 05 2020
a(n) ~ 2^(-9/8) * 3^(1/8) * n^(n - 1/8) * exp(1/54 - n^(1/4)/(2^(15/4)*3^(5/4)) - sqrt(6*n)/12 + 2^(7/4)*3^(-3/4)*n^(3/4) - n). - Vaclav Kotesovec, Jun 11 2020
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n+k-1,n-2*k)/(2^k * k!). - Seiichi Manyama, Jun 17 2024

A361572 Expansion of e.g.f. exp( (x / (1-x))^3 ).

Original entry on oeis.org

1, 0, 0, 6, 72, 720, 7560, 90720, 1270080, 20381760, 364694400, 7125148800, 150186960000, 3393726336000, 81882210009600, 2102315389574400, 57244753133568000, 1647544166940672000, 49957730917981286400, 1591303422125646028800

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A000262, A052887, A361576.

Cf. A091695.

Table[n! * Sum[Binomial[n-1,n-3*k]/k!, {k,0,n/3}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)
With[{nn=20},CoefficientList[Series[Exp[(x/(1-x))^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 07 2024 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((x/(1-x))^3)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, (-1)^(j-3)*j*binomial(-3, j-3)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-1,n-3*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} (-1)^(k-3) * k * binomial(-3,k-3) * a(n-k)/(n-k)!.
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = 4*(n-1)*a(n-1) - 6*(n-2)*(n-1)*a(n-2) + (n-2)*(n-1)*(4*n - 9)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(1/8) * exp(-1/4 + 5*3^(-1/4)*n^(1/4)/8 - sqrt(3*n)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n - 1/8) / 2 * (1 - (409/2560)*3^(1/4)/n^(1/4)). (End)

A364940 E.g.f. satisfies A(x) = exp( xA(x) / (1 - xA(x))^3 ).

Original entry on oeis.org

1, 1, 9, 124, 2525, 68616, 2338357, 96004672, 4616135001, 254542038400, 15839013320801, 1098078537291264, 83940831427695541, 7014958697801657344, 636298582947212386125, 62261039244978489081856, 6537251350698278868150833, 733159568772947522820538368

Offset: 0

Seiichi Manyama, Aug 14 2023

nonn

Index entries for reversions of series

Cf. A052873, A364939.

Cf. A091695.

a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(n+2*k-1, n-k)/k!);

a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(n+2*k-1,n-k)/k!.

E.g.f.: (1/x) * Series_Reversion( x*exp(-x/(1 - x)^3) ). - Seiichi Manyama, Sep 23 2024

A387244 Expansion of e.g.f. exp(x^2/(1-x)^4).

Original entry on oeis.org

1, 0, 2, 24, 252, 2880, 38280, 594720, 10565520, 209502720, 4558407840, 107702179200, 2744400415680, 75016089308160, 2189152249764480, 67906418407027200, 2230160988344889600, 77271779968704921600, 2815893910009609228800, 107629691727791474841600, 4304364116456244429388800

Offset: 0

Vaclav Kotesovec, Aug 24 2025

In general, if s >= 1, 1 <= r <= s and e.g.f. = exp(x^r/(1-x)^s) then for n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + (s-r)*k - 1, s*k - 1)/k!.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A293012, A000262, A082579, A091695, A361283.

Cf. A052845, A052887, A386514.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x^2/(1-x)^4))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Aug 25 2025

nmax=20; CoefficientList[Series[E^(x^2/(1-x)^4), {x, 0, nmax}], x] * Range[0, nmax]!
nmax=20; Join[{1}, Table[n!*Sum[Binomial[n+2*k-1, 4*k-1]/k!, {k, 1, n}], {n, 1, nmax}]]
Join[{1}, Table[n!*n*(n - 1)*(n + 1)/6 * HypergeometricPFQ[{1 - n/2, 3/2 - n/2, 1 + n/2, 3/2 + n/2}, {5/4, 3/2, 7/4, 2}, 1/16], {n, 1, 20}]]

For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n + 2*k - 1, 4*k - 1)/k!.
a(n) = 5*(n-1)*a(n-1) - 2*(n-1)*(5*n-11)*a(n-2) + 2*(n-2)*(n-1)*(5*n-14)*a(n-3) - 5*(n-4)*(n-3)*(n-2)*(n-1)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ 2^(1/5) * 5^(-1/2) * exp(1/80 - 2^(-9/5)*n^(2/5)/3 + 5*2^(-8/5)*n^(4/5) - n) * n^(n - 1/10).

cp's OEIS Frontend

A293012 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1 - x)^k).

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A361279 Expansion of e.g.f. exp(x * (1+x)^3).

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A380515 Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A361599 Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x).

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A367789 E.g.f. satisfies A(x) = exp( x/(1-x)^3 * A(x) ).

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A250917 Expansion of e.g.f. exp( x*C(x)^3 ) where C(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers, A000108.

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A335345 Expansion of e.g.f. exp(x^2/(2*(1 - x)^3)).

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Mathematica

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A361572 Expansion of e.g.f. exp( (x / (1-x))^3 ).

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A380515 Expansion of e.g.f. exp(xG(x)^3) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

A250917 Expansion of e.g.f. exp( xC(x)^3 ) where C(x) = (1 - sqrt(1-4x))/(2*x) is the g.f. of the Catalan numbers, A000108.

A364940 E.g.f. satisfies A(x) = exp( xA(x) / (1 - xA(x))^3 ).