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.

Showing 1-10 of 10 results.

A156326 E.g.f.: A(x) = exp( Sum_{n>=1} n^2 * a(n-1)*x^n/n! ) = Sum_{n>=0} a(n)*x^n/n! with a(0) = 1.

Original entry on oeis.org

1, 1, 5, 58, 1181, 36696, 1601497, 92969920, 6908883417, 638746871680, 71860612355981, 9664570175364864, 1531263494465900725, 282321785979644121088, 59935663751282958139425, 14517627118656645274771456, 3980008380007702720451029553, 1226189930561023692489563013120
Offset: 0

Views

Author

Paul D. Hanna, Feb 08 2009

Keywords

Examples

			E.g.f: A(x) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! + ...
log(A(x)) = x + 2^2*x^2/2! + 3^2*5*x^3/3! + 4^2*58*x^4/4! + 5^2*1181*x^5/5! + ...

Links

Crossrefs

Cf. A156325, A156327, A182962, A161968.

Programs

  • Mathematica
    nmax = 20; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = Sum[k^2 * Binomial[n-1,k-1]*b[[k]]*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b (* Vaclav Kotesovec, Feb 27 2014 *)
  • PARI
    {a(n)=if(n==0,1,n!*polcoeff(exp(sum(k=1,n,k^2*a(k-1)*x^k/k!)+x*O(x^n)),n))}
  • PARI
    {a(n)=if(n==0,1,sum(k=1,n,k^2*binomial(n-1,k-1)*a(k-1)*a(n-k)))}
  • PARI
    seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n] = sum(k=1, n, k^2 * binomial(n-1,k-1)*a[k]*a[1+n-k])); a} \\ Andrew Howroyd, Jan 05 2020

Formula

a(n) = Sum_{k=1..n} k^2 * C(n-1,k-1)*a(k-1)*a(n-k) for n>0, with a(0)=1.
E.g.f.: A(x) = exp( x*A(x) + x^2*A'(x) ). - Paul D. Hanna, Apr 02 2018
E.g.f.: A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x/G(x)) = G(x) is the e.g.f. of A182962, which satisfies:
. G(x) = exp( x/(1 - x*G'(x)/G(x)) );
. a(n) = [x^n/n!] G(x)^(n+1)/(n+1) for n>=0.
a(n) = A161968(n+1)/(n+1), where L(x) = x*exp(x*d/dx L(x)) is the e.g.f. of A161968. - Paul D. Hanna, Feb 21 2014
a(n) ~ c * n * (n!)^2, where c = A238223 * exp(1) = 0.592451670452494179138706062417512405957... - Vaclav Kotesovec, Feb 27 2014

Extensions

Terms a(15) and beyond from Andrew Howroyd, Jan 05 2020

A385946 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+4,4) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 6, 106, 4176, 316696, 42104392, 9172761368, 3106804304704, 1567537597699840, 1137145604406018176, 1151190083860345401984, 1585522852991230263395584, 2906652632758146061798315776, 6959140466024956612239458880000, 21400639132670591710876896798678016
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A000272, A156325, A385945, A385947, A385948.
Cf. A385953.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, binomial(j+4, 4)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( Sum_{k=0..3} binomial(3,k) * x^(k+1)/(k+1)! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385945 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+3,3) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 5, 63, 1533, 62736, 3969387, 366744330, 47441881377, 8313978813120, 1921417594566561, 572533956456137424, 215766174031503450885, 101144655173329674617088, 58127411808811103704523775, 40435528907318329027426583376, 33666103690446265067517343384833
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A000272, A156325, A385946, A385947, A385948.
Cf. A385952.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, binomial(j+3, 3)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( Sum_{k=0..2} binomial(2,k) * x^(k+1)/(k+1)! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385947 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+5,5) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 7, 166, 10029, 1321025, 341733205, 160453080950, 128422430092385, 166469443066352440, 334968718604910165425, 1009644894131844004090200, 4422360688027934597152329025, 27423466157672001507611296316100, 235350249980804930971638499216115775
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A000272, A156325, A385945, A385946, A385948.
Cf. A385954.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, binomial(j+5, 5)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( Sum_{k=0..4} binomial(4,k) * x^(k+1)/(k+1)! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385948 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+6,6) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 8, 246, 21750, 4689546, 2197062708, 2046202234224, 3528088593902364, 10627093734265740672, 53295889303479275834616, 427383379745842299684115608, 5294446934064450139154214169992, 98355143996083993836475641916586304, 2669951662594756888115675117287929721248
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A000272, A156325, A385945, A385946, A385947.
Cf. A385955.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, binomial(j+6, 6)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( Sum_{k=0..5} binomial(5,k) * x^(k+1)/(k+1)! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385101 E.g.f. A(x) satisfies A(x) = exp(x * A(x) + x^4/24 * A'''(x)).

Original entry on oeis.org

1, 1, 3, 16, 141, 2161, 59842, 2979509, 258264379, 37321303420, 8597483041421, 3028595626839564, 1572449537786394577, 1165432782899826271026, 1199378312656505145280950, 1673258190849282722438631406, 3099020844849243071430739707913, 7481267275389054589164201426886656
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A156325, A386533, A386534.
Cf. A385921.

Programs

  • Mathematica
    terms = 18; A[] = 1; Do[A[x] = Exp[x*A[x]+x^4*A'''[x]/24] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] * Range[0,terms-1]! (* Stefano Spezia, Aug 04 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(1+sum(k=1, 3, stirling(3, k, 1)*j^k)/24)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + (2*k - 3*k^2 + k^3)/24) * binomial(n-1,k) * a(k) * a(n-1-k).

A386533 E.g.f. A(x) satisfies A(x) = exp(x * A(x) + x^3/6 * A''(x)).

Original entry on oeis.org

1, 1, 3, 19, 225, 4576, 149517, 7448134, 542269961, 55702422400, 7832607617351, 1468762340728464, 359026336711386577, 112153290859090469184, 44001791667365123420025, 21354097196759712722857776, 12647439446531876144344860113, 9033421564454672567830839315456
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A156325, A385101, A386534.
Cf. A385920.

Programs

  • Mathematica
    terms = 18; A[] = 1; Do[A[x] = Exp[x*A[x]+ x^3A''[x]/6] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Aug 04 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(1+sum(k=1, 2, stirling(2, k, 1)*j^k)/6)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + (-k + k^2)/6) * binomial(n-1,k) * a(k) * a(n-1-k).

A386534 E.g.f. A(x) satisfies A(x) = exp(x * A(x) + x^5/120 * A''''(x)).

Original entry on oeis.org

1, 1, 3, 16, 125, 1421, 26833, 968626, 70638465, 10215072856, 2782227253373, 1347216023489436, 1099522113403916545, 1443781044602756539876, 2930977624516859360997387, 8889808786962394898290294048, 39115513670641030174644662148305, 243377943140592361750259305827057888
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A156325, A385101, A386533.
Cf. A385922.

Programs

  • Mathematica
    terms = 18; A[] = 1; Do[A[x] = Exp[x*A[x]+x^5*A''''[x]/120] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Aug 04 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(1+sum(k=1, 4, stirling(4, k, 1)*j^k)/120)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + (-6*k + 11*k^2 - 6*k^3 + k^4)/120) * binomial(n-1,k) * a(k) * a(n-1-k).

A156327 E.g.f.: A(x) = exp( Sum_{n>=1} n*(n+3)/2 * a(n-1)*x^n/n! ) = Sum_{n>=0} a(n)*x^n/n! with a(0)=1.

Original entry on oeis.org

1, 2, 14, 194, 4280, 134232, 5587408, 294882464, 19102334112, 1482726089600, 135370060595264, 14325189014356992, 1736329123715436544, 238698935851482530816, 36911830664814417907200, 6375425555384677316100608, 1222423907917065757088181248, 258802786174190320917263867904
Offset: 0

Views

Author

Paul D. Hanna, Feb 08 2009

Keywords

Examples

			E.g.f: A(x) = 1 + 2*x + 14*x^2/2! + 194*x^3/3! + 4280*x^4/4! + 134232*x^5/5! +...
log(A(x)) = 2*1*x + 5*2*x^2/2! + 9*14*x^3/3! + 14*194*x^4/4! + 20*4280*x^5/5! +...

Crossrefs

Cf. A156325, A156326.

Programs

  • Mathematica
    terms = 18; A[] = 1; Do[A[x] = Exp[2x*A[x]+x^2*A'[x]/2] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] * Range[0,terms-1]! (* Stefano Spezia, Aug 04 2025 *)
  • PARI
    {a(n)=if(n==0,1,n!*polcoeff(exp(sum(k=1,n,k*(k+3)/2*a(k-1)*x^k/k!)+x*O(x^n)),n))}
  • PARI
    {a(n)=if(n==0,1,sum(k=1,n,k*(k+3)/2*binomial(n-1,k-1)*a(k-1)*a(n-k)))}
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(2+j/2)*binomial(i-1, j)*v[j+1]*v[i-j])); v; \\ Seiichi Manyama, Jul 25 2025

Formula

a(n) = Sum_{k=1..n} k*(k+3)/2 * C(n-1,k-1)*a(k-1)*a(n-k) for n>0, with a(0)=1.
E.g.f. A(x) satisfies A(x) = exp(2 * x * A(x) + x^2/2 * A'(x)). - Seiichi Manyama, Jul 25 2025
a(n) ~ c * n!^2 * n^7 / 2^n, where c = 0.00029014625163457216349268... - Vaclav Kotesovec, Aug 05 2025

A385322 E.g.f. A(x) satisfies A(x) = exp(x + x^2/2 * A'(x)).

Original entry on oeis.org

1, 1, 2, 10, 94, 1386, 29146, 822928, 29927612, 1359897724, 75429391276, 5013317213136, 393252908602720, 35947138247529952, 3787896504780614864, 455830291523748797776, 62132218311858089750416, 9523376536749853754046864, 1630878867943272798422352400, 310238352508716702644975872192
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A386531, A386532.
Cf. A143925, A156325.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, (1+j)*j*binomial(i-1, j)*v[j+1]*v[i-j])/2); v;

Formula

a(0) = 1; a(n) = a(n-1) + (1/2) * Sum_{k=0..n-1} (1 + k) * k * binomial(n-1,k) * a(k) * a(n-1-k).
Showing 1-10 of 10 results.

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