A007558 -id:A007558 - cp's OEIS frontend

A035012 One half of 9-factorial numbers.

1, 11, 220, 6380, 242440, 11394680, 638102080, 41476635200, 3069271004800, 254749493398400, 23436953392652800, 2367132292657932800, 260384552192372608000, 30985761710892340352000, 3966177498994219565056000, 543366317362208080412672000, 79331482334882379740250112000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..325
Index entries for sequences related to factorial numbers.

Cf. A007558, A034171, A035012, A035013, A035017, A035018.

Cf. A035020, A035021, A035022, A035023, A045756, A084949.

[n le 1 select 1 else (9*n-7)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 18 2022

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 10, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Table[9^n*Pochhammer[2/9, n]/2, {n, 40}] (* G. C. Greubel, Oct 18 2022 *)

[9^n*rising_factorial(2/9,n)/2 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

2*a(n) = (9*n-7)(!^9) := Product_{j=1..n} (9*j - 7).
E.g.f.: (-1+(1-9*x)^(-2/9))/2.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/2) * 9^n * Pochhammer(n, 2/9).
a(n) = (9*n-7)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A084949(n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/9^7)^(1/9)*(Gamma(2/9) - Gamma(2/9, 1/9)). (End)

A063902 a(n+1) = sum{j = 0,...n}[C(2n,2j)a(j)a(n-j)] with a(0) = 1.

Original entry on oeis.org

1, 1, 2, 10, 80, 1000, 17600, 418000, 12848000, 496672000, 23576960000, 1348404640000, 91442700800000, 7255463564800000, 665885747225600000, 69994087116448000000, 8354181454767104000000, 1123646013779238400000000, 169165728883243642880000000

Offset: 0

Henry Bottomley, Aug 30 2001

nonn

Define e.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!. Then A(x) = P(z + x/sqrt(6)) where P is the Weierstrass P-function with g_2 = 0, g_3 = 4 and z = 2.10327313773966...*i satisfies P(z) = 1, P'(z) = 0. - Michael Somos, Apr 18 2022

			a(3) = 1*a(0)a(2) + 6*a(1)a(1) + 1*a(2)a(0) = 2+6+2 = 10.
E.g.f.: A(x) = 1 + x^2/2! + 2*x^4/4! + 10*x^6/6! + 80*x^8/8! + ...
G.f. = 1 + x + 2*x^2 + 10*x^3 + 80*x^4 + 1000*x^8 + 17600*x^9 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..270
Markus Kuba, Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.

a(n+1) = sum[C(n, j)a(j)a(n-j)] would give factorials A000142, a(n+1) = sum[C(2n, 2j)a(j)a(n-j)]/a(n) would give Catalan numbers A000108, a(n+1) = sum[C(n, j)a(j)a(n-j)]/a(n) would give central binomials A001405.

Cf. A007558, A258895.

Clear[a]; a[n_]:=a[n]=If[n<2,1,Sum[a[j]*a[n-1-j]*Binomial[2*n-2,2*j],{j,0,n-1}]]; Table[a[n], {n,0,30}] (* Vaclav Kotesovec, Jun 14 2015 *)
a[ n_] := If[n < 0, 0, Module[{x, y}, Nest[Dt[#] /. {Dt[x] -> y, Dt[y] -> x^2} &, x, 2*n] /. {x -> 1, y -> 0}]]; (* Michael Somos, Apr 18 2022 *)

/* E.g.f. A(x) = exp( Integral 1/A(x) * Integral A(x)^2 dx dx ) */
{a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( 1/A * intformal( A^2 + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
for(n=0,20,print1(a(2*n),", ")) \\ Paul D. Hanna, Jun 02 2015

/* By definition: */
{a(n) = if(n==0,1,sum(k=0,n-1, binomial(2*n-2,2*k)*a(k)*a(n-k-1)))}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Jun 02 2015

{a(n) = if(n<0, 0, my(x='x, y = 1 + O(x^2)); for(i=1, n, y = 1 + intformal(intformal(y^2))); polcoeff(y, 2*n)*(2*n)!)}; /* Michael Somos, Apr 18 2022 */

E.g.f. satisfies: A(x) = exp( Integral 1/A(x) * Integral A(x)^2 dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(1*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, Jun 02 2015
From Vaclav Kotesovec, Jun 14 2015: (Start)
a(n) ~ c * d^n * n!^2 * sqrt(n), where d = A258895 = 32*Pi / (Gamma(1/6) * Gamma(1/3))^2 = 2^(17/3) * Pi^2 / (3*Gamma(1/3)^6) = 0.452104299183420528841..., c = 1.53043521765866544548745...  = 2^(20/3) * Pi^(3/2) / Gamma(1/3)^6 = 192*sqrt(Pi) / (Gamma(1/3)*Gamma(1/6))^2.
a(n) ~ 3 * 2^(5*n+7) * Pi^(n+3/2) * n^(2*n+3/2) / (exp(2*n) * Gamma(1/6)^(2*n+2) * Gamma(1/3)^(2*n+2)).
a(n) ~ 2^((17*n+23)/3) * Pi^(2*n+5/2) * n^(2*n+3/2) / (3^n * exp(2*n) * Gamma(1/3)^(6*n+6)).
(End)
Define e.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!. Then 3*A'(x)^2 = 2*A(x)^3 - 2. - Michael Somos, Jan 07 2022

More terms from Vaclav Kotesovec, Jun 14 2015

A333497 a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * a(k) * a(n-k-3).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 18, 48, 144, 456, 1560, 5808, 23184, 98160, 440832, 2101824, 10588608, 56104128, 312013440, 1818498816, 11082682368, 70467474816, 466680045312, 3214497245184, 22994283345408, 170573216656896, 1310482565462016, 10415453732637696

Offset: 0

Ilya Gutkovskiy, Jul 04 2020

nonn

Shifts 3 places left when e.g.f. is squared.

Georg Fischer, Table of n, a(n) for n = 0..500

Cf. A000142, A007558, A307970, A336009, A336010.

a[0] = a[1] = a[2] = 1; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 28}]
nmax = 28; A[] = 0; Do[A[x] = 1 + x + x^2/2 + Integrate[Integrate[Integrate[A[x]^2, x], x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!

E.g.f. A(x) satisfies: A(x) = 1 + x + x^2/2 + Integral( Integral( Integral A(x)^2 dx) dx) dx.

A336009 a(0) = ... = a(3) = 1; a(n) = Sum_{k=0..n-4} binomial(n-4,k) * a(k) * a(n-k-4).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 8, 16, 34, 82, 226, 678, 2108, 6892, 23948, 88532, 344816, 1401200, 5925000, 26146360, 120743496, 582606552, 2926675112, 15259183112, 82458502624, 461577781968, 2674216518016, 16013654472352, 98968416103968, 630595248710144

Offset: 0

Ilya Gutkovskiy, Jul 04 2020

nonn

Shifts 4 places left when e.g.f. is squared.

Cf. A000142, A007558, A307971, A333497, A336010.

a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = Sum[Binomial[n - 4, k] a[k] a[n - k - 4], {k, 0, n - 4}]; Table[a[n], {n, 0, 30}]
nmax = 30; A[] = 0; Do[A[x] = 1 + x + x^2/2 + x^3/6 + Integrate[Integrate[Integrate[Integrate[A[x]^2, x], x], x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!

E.g.f. A(x) satisfies: A(x) = 1 + x + x^2/2 + x^3/6 + Integral( Integral( Integral( Integral A(x)^2 dx) dx) dx) dx.

A336010 a(0) = ... = a(4) = 1; a(n) = Sum_{k=0..n-5} binomial(n-5,k) * a(k) * a(n-k-5).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 66, 148, 374, 1052, 3156, 9724, 31096, 104124, 366696, 1355624, 5220120, 20763160, 84944720, 357759200, 1557192440, 7029575320, 32929457880, 159764303320, 800509163360, 4132518624560, 21953331512080, 119966645509440

Offset: 0

Ilya Gutkovskiy, Jul 04 2020

nonn

Shifts 5 places left when e.g.f. is squared.

Cf. A000142, A007558, A307972, A333497, A336009.

a[0] = a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = Sum[Binomial[n - 5, k] a[k] a[n - k - 5], {k, 0, n - 5}]; Table[a[n], {n, 0, 32}]
nmax = 32; A[] = 0; Do[A[x] = 1 + x + x^2/2 + x^3/6 + x^4/24 + Integrate[Integrate[Integrate[Integrate[Integrate[A[x]^2, x], x], x], x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!

E.g.f. A(x) satisfies: A(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + Integral( Integral( Integral( Integral( Integral A(x)^2 dx) dx) dx) dx) dx.

A336023 a(0) = a(1) = a(2) = 1; a(n) = (1/2) * Sum_{k=0..n-2} binomial(n-2,k) * a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 9, 24, 73, 241, 881, 3526, 15252, 70989, 354454, 1887349, 10674773, 63936181, 404240736, 2690174976, 18798077227, 137612318374, 1053181689299, 8410678868749, 69966341141373, 605313810207096, 5438284743262816, 50668461879077851

Offset: 0

Ilya Gutkovskiy, Jul 05 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A000111, A007558.

a[0] = a[1] = a[2] = 1; a[n_] := a[n] = (1/2) Sum[Binomial[n - 2, k] a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 27}]
nmax = 27; A[] = 0; Do[A[x] = 1 + x + x^2/4 + 1/2 Integrate[Integrate[A[x]^2, x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!
Table[SeriesCoefficient[1 + (72 (WeierstrassP[x, {-1/12, 1/432}] - WeierstrassPPrime[x, {-1/12, 1/432}]))/(1 - 12 WeierstrassP[x, {-1/12, 1/432}])^2, {x, 0, k}] k!, {k, 0, 30}] (* Jan Mangaldan, Nov 27 2020 *)
a[ n_] := Module[{b, b0, b1, b2}, b[0]=b0; b[m_] := b[n] = Expand[Dt[b[m-1]] /. {Dt[b0]->b1, Dt[b1]->b2, Dt[b2]->b0*b1}]; b[n] /. {b0->1, b1->1, b2->1}]; (* Michael Somos, Jan 21 2021 *)

{a(n) = my(t, b0='b0, b1='b1, b2='b2); t = b0; for(k=1, n, t = deriv(t, b0) * b1 + deriv(t, b1) * b2 + deriv(t, b2) * b0*b1); substvec(t, [b0, b1, b2], [1, 1, 1])}; /* Michael Somos, Jan 21 2021 */

E.g.f. A(x) satisfies: A(x) = 1 + x + x^2/4 + (1/2) * Integral( Integral A(x)^2 dx) dx.
a(n) ~ n! * c * d^n * n, where d = 0.3327497845052484431663758951399704951054382840130942017786917698589406... is the root of the equation WeierstrassP(1/d, -1/12, 1/432) = 1/12 and c = 1.3286690290594713229820603959597228439699874045711069972066245497... - Vaclav Kotesovec, Jul 05 2020, updated Nov 28 2020
E.g.f.: 12^(1/3) * WeierstrassP((x + c)/12^(1/3), {-12^(1/3), 1/3}), where c = 6.010522299732561104963296256980218197216667207108031458739075310344... is the root of the equation WeierstrassP(c/12^(1/3), {-12^(1/3), 1/3}) = 12^(-1/3). - Vaclav Kotesovec, Jul 05 2020

A337187 a(n) = 1 + Sum_{k=0..n-2} binomial(n-2,k) * a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, 2, 3, 7, 19, 63, 229, 955, 4407, 22445, 124249, 746003, 4821287, 33394193, 246652725, 1935828995, 16086138151, 141100295557, 1302780182449, 12630092274099, 128275445380247, 1362029496267529, 15090795795916493, 174167341456580947, 2090520625244752407

Offset: 0

Ilya Gutkovskiy, Jan 29 2021

nonn

Cf. A007558, A054687, A090344.

a[n_] := a[n] = 1 + Sum[Binomial[n - 2, k] a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]

E.g.f. A(x) satisfies: A''(x) = exp(x) + A(x)^2.

A351879 a(0) = a(1) = 1; a(n) = -Sum_{k=0..n-2} binomial(n-2,k) * a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, -1, -2, 0, 10, 10, -60, -220, 400, 4200, 2200, -90200, -290400, 1892000, 15796000, -24024000, -775676000, -1592492000, 36509880000, 240055640000, -1435950560000, -23703057840000, 7376731120000, 2082346354000000, 9478853472000000, -162472029808000000

Offset: 0

Ilya Gutkovskiy, Feb 23 2022

sign

Cf. A007558, A307374, A346078.

a[0] = a[1] = 1; a[n_] := a[n] = -Sum[Binomial[n - 2, k] a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 26}]

E.g.f. A(x) satisfies: A(x) = 1 + x - Integral( Integral A(x)^2 dx) dx.

cp's OEIS Frontend

A035012 One half of 9-factorial numbers.

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A063902 a(n+1) = sum{j = 0,...n}[C(2n,2j)a(j)a(n-j)] with a(0) = 1.

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A333497 a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * a(k) * a(n-k-3).

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A336009 a(0) = ... = a(3) = 1; a(n) = Sum_{k=0..n-4} binomial(n-4,k) * a(k) * a(n-k-4).

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A336010 a(0) = ... = a(4) = 1; a(n) = Sum_{k=0..n-5} binomial(n-5,k) * a(k) * a(n-k-5).

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A336023 a(0) = a(1) = a(2) = 1; a(n) = (1/2) * Sum_{k=0..n-2} binomial(n-2,k) * a(k) * a(n-k-2).

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A337187 a(n) = 1 + Sum_{k=0..n-2} binomial(n-2,k) * a(k) * a(n-k-2).

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A351879 a(0) = a(1) = 1; a(n) = -Sum_{k=0..n-2} binomial(n-2,k) * a(k) * a(n-k-2).

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