A143916 -id:A143916 - cp's OEIS frontend

A143917 G.f. A(x) satisfies A(x) = 1/(1-x) + x^2A(x)A'(x).

1, 1, 2, 6, 25, 133, 851, 6313, 53061, 497493, 5144500, 58161126, 713789847, 9453038227, 134405493652, 2042529150110, 33045300698761, 567165849906233, 10294218618819268, 197022941365579804, 3966001076798967837, 83767346751954718361, 1852440991624711835677

Offset: 0

Paul D. Hanna, Sep 05 2008

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 133*x^5 + 851*x^6 +...
A'(x) = 1 + 4*x + 18*x^2 + 100*x^3 + 665*x^4 + 5106*x^5 +...
A(x)*A'(x) = 1 + 5*x + 24*x^2 + 132*x^3 + 850*x^4 + 6312*x^5 +...

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

Cf. A143916 (variant), A238214.

Clear[a]; a[0] = 1; a[n_]/; n>=1 := a[n] = 1 + Sum[(k - 1) a[k - 1] a[n - k], {k, n}]; Table[a[n], {n,0, 16}] (* David Callan, Jun 24 2013 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x+x*O(x^n))+x^2*A*deriv(A)); polcoeff(A, n)}

a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, j*v[j+1]*v[i-j])); v; \\ Seiichi Manyama, Jul 10 2025

a(n) ~ c * n!, where c = 1.81857005675331400362707139219522893237... (see A238214). - Vaclav Kotesovec, Feb 20 2014
a(n) = 1 + Sum_{k=0..n-1} k * a(k) * a(n-1-k). - Seiichi Manyama, Jul 10 2025
a(n) = 1 + (n-1)/2 * Sum_{k=0..n-1} a(k) * a(n-1-k). - Seiichi Manyama, Jul 15 2025

A185183 G.f. A(x) satisfies A(x) = 1+x + x^2*[d/dx A(x)^2].

Original entry on oeis.org

1, 1, 2, 10, 72, 672, 7640, 102072, 1564864, 27064448, 521248320, 11064781760, 256702399360, 6462978471168, 175520877380992, 5115062135795584, 159227683153536000, 5273353734210310144, 185143079148664099840, 6869062513111759635456

Offset: 0

Paul D. Hanna, Mar 12 2012

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 72*x^4 + 672*x^5 + 7640*x^6 +...
Related series:
A(x)^2 = 1 + 2*x + 5*x^2 + 24*x^3 + 168*x^4 + 1528*x^5 + 17012*x^6 +...
d/dx A(x)^2 = 2 + 10*x + 72*x^2 + 672*x^3 + 7640*x^4 + 102072*x^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A143916, A218223, A218224, A376125.

{a(n)=local(A=1+x); for(i=1, n, A=1+x+x^2*deriv(A^2+x*O(x^n))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies: A(x) = 1+x + 2*x^2*A(x)*A'(x).
a(n) ~ c * n! * 2^n / sqrt(n), where c = 0.493602095524198015213766719826126125048... - Vaclav Kotesovec, Feb 21 2014
a(0) = a(1) = 1; a(n) = (n-1) * Sum_{k=0..n-1} a(k) * a(n-k-1). - Ilya Gutkovskiy, Jul 05 2020
a(0) = a(1) = 1; a(n) = 2 * Sum_{k=0..n-1} k * a(k) * a(n-1-k). - Seiichi Manyama, Jul 16 2025

A238214 Decimal expansion of a constant related to A143917.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Feb 21 2014

			1.81857005675331400362707139219522893237...

Cf. A143917, A143916.

Equals lim n->infinity A143917(n)/n!.

A217989 G.f. satisfies: A(x) = 1+x + x^2A'(x)A(x)^2.

Original entry on oeis.org

1, 1, 1, 4, 19, 116, 835, 6890, 63826, 654552, 7354893, 89830770, 1184915556, 16788863356, 254342837905, 4103256660048, 70241858430220, 1271839899568064, 24287699718766932, 487891841580468294, 10285169201486942788, 227042177973572054900

Offset: 0

Paul D. Hanna, Oct 23 2012

nonn

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 19*x^4 + 116*x^5 + 835*x^6 + 6890*x^7 +...
Related expansions:
A'(x) = 1 + 2*x + 12*x^2 + 76*x^3 + 580*x^4 + 5010*x^5 + 48230*x^6 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 10*x^3 + 47*x^4 + 278*x^5 + 1956*x^6 + 15834*x^7 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..445

Cf. A143916, A218223.

{a(n)=local(A=1+x); for(i=1, n, A=1+x+x^2*A'*(A^2+x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

a(n) ~ c * n! * n, where c = 0.21362630601338471861707529847387... - Vaclav Kotesovec, Feb 22 2014

cp's OEIS Frontend

A143917 G.f. A(x) satisfies A(x) = 1/(1-x) + x^2A(x)A'(x).

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A185183 G.f. A(x) satisfies A(x) = 1+x + x^2*[d/dx A(x)^2].

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A238214 Decimal expansion of a constant related to A143917.

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A217989 G.f. satisfies: A(x) = 1+x + x^2A'(x)A(x)^2.

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cp's OEIS Frontend

A143917 G.f. A(x) satisfies A(x) = 1/(1-x) + x^2*A(x)*A'(x).

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A185183 G.f. A(x) satisfies A(x) = 1+x + x^2*[d/dx A(x)^2].

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Author

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Examples

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PARI

Formula

A238214 Decimal expansion of a constant related to A143917.

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Author

Keywords

Examples

Crossrefs

Formula

A217989 G.f. satisfies: A(x) = 1+x + x^2*A'(x)*A(x)^2.

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A143917 G.f. A(x) satisfies A(x) = 1/(1-x) + x^2A(x)A'(x).

A217989 G.f. satisfies: A(x) = 1+x + x^2A'(x)A(x)^2.