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.

Previous Showing 11-19 of 19 results.

A385942 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + k^5) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 5, 508, 497861, 2554041696, 47918955042217, 2608995595530944320, 350836859825187730934697, 103472315352121087796983183360, 61101436986101317921145771113951181, 67212924933426575369862458525709786073344, 129898118403746997254471428114728554653243564525
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A156326, A385939, A385940, A385941, A385943.
Cf. A385833.

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, (1+j)*(1+j^5)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( x*A(x) + x*Sum_{k=1..5} Stirling2(5,k) * x^k * (d^k/dx^k A(x)) ).

A385943 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + k^6) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 5, 988, 2888933, 59194266336, 5550172939486537, 1812719786900514856960, 1706146365658760367161728617, 4025335006744077207541517795929600, 21392361120121469487882204135345762936461, 235316442953945260569915546964215106936729204224
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A156326, A385939, A385940, A385941, A385942.
Cf. A385834.

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, (1+j)*(1+j^6)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( x*A(x) + x*Sum_{k=1..6} Stirling2(6,k) * x^k * (d^k/dx^k A(x)) ).

A385923 E.g.f. A(x) satisfies A(x) = exp(x*A(x) + x^6*A'''''(x)).

Original entry on oeis.org

1, 1, 3, 16, 125, 1296, 949927, 4800957904, 96864153387129, 5860087724767012480, 886162470100464297115691, 294792579950929452096468136704, 196126682670165049397384798842463797, 242323538289386581241948100813652397771776, 523949046624700150687300336366625589891821933775
Offset: 0

Views

Author

Seiichi Manyama, Jul 12 2025

Keywords

Crossrefs

Cf. A000272, A156326, A385920, A385921, A385922.
Cf. A385765.

Programs

  • Mathematica
    terms = 15; A[] = 1; Do[A[x] = Exp[x*A[x]+x^6*A'''''[x]] + 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, 5, stirling(5, k, 1)*j^k))*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 + 24*k - 50*k^2 + 35*k^3 - 10*k^4 + k^5) * binomial(n-1,k) * a(k) * a(n-1-k).

A161968 E.g.f. L(x) satisfies: L(x) = x*exp(x*d/dx L(x)), where L(x) is the logarithm of e.g.f. of A161967.

Original entry on oeis.org

1, 2, 15, 232, 5905, 220176, 11210479, 743759360, 62179950753, 6387468716800, 790466735915791, 115974842104378368, 19906425428056709425, 3952505003715017695232, 899034956269244372091375, 232282033898506324396343296, 67660142460130946247667502401
Offset: 1

Views

Author

Paul D. Hanna, Jun 23 2009

Keywords

Examples

			E.g.f.: L(x) = x + 2*x^2/2! + 15*x^3/3! + 232*x^4/4! + 5905*x^5/5! +...
where exp(L(x)) = exp(x*exp(x*L'(x))) = e.g.f. of A161967:
exp(L(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 317*x^4/4! + 7596*x^5/5! +...
and exp(x*L'(x)) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! +...+ A156326(n)*x^n/n! +...
RELATED EXPRESSIONS.
E.g.f.: A(x) = 1 + 2*x + 15*x^2/2! + 232*x^3/3! + 5905*x^4/4! +...
where
A(x) = d/dx x*exp(x*A(x)) = exp(x*A(x)) * (1 + x*A(x) + x^2*A'(x)) with
exp(x*A(x)) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! + 1601497*x^6/6! + 92969920*x^7/7! +...+ A156326(n)*x^n/n! +...

Crossrefs

Cf. A161967 (exp), A156326.

Programs

  • PARI
    {a(n)=local(L=x+x^2);for(i=1,n,L=x*exp(x*deriv(L)+O(x^n)));n!*polcoeff(L,n)}
for(n=1,30,print1(a(n),", "))

Formula

a(n) = n * A156326(n-1), where the e.g.f. of A156326 satisfies: Sum_{n>=0} A156326(n)*x^n/n! = exp( Sum_{n>=1} n^2 * A156326(n-1)*x^n/n! ) = exp( Sum_{n>=1} n * a(n)*x^n/n! ). - Paul D. Hanna, Feb 21 2014
E.g.f. A(x), with offset=0, satisfies [Paul D. Hanna, Feb 15 2015]:
(1) A(x) = d/dx x*exp(x*A(x)).
(2) A(x) = exp(x*A(x)) * (1 + x*A(x) + x^2*A'(x)).
(3) exp(x*A(x)) = e.g.f. of A156326.

A385979 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * binomial(k+2,2) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 7, 145, 6449, 522096, 69506737, 14186121706, 4212887224905, 1747635451186240, 979909591959562571, 722787600597422326704, 685585597413868516073953, 820283211774547803576454720, 1217648676024408903145299884925, 2210504358495882876855897821031376
Offset: 0

Views

Author

Seiichi Manyama, Jul 14 2025

Keywords

Crossrefs

Cf. A000272, A156326, A385980, A385981, A385982.

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, (j+1)*binomial(j+2, 2)*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! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385980 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * binomial(k+3,3) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 9, 295, 24921, 4504516, 1543745107, 919392117722, 890353538984905, 1330464112593541120, 2940642877993896450701, 9284167814032856189142864, 40666099850492306669400356041, 241073945237343019120798232332320, 1893421587381601800604423881821405775
Offset: 0

Views

Author

Seiichi Manyama, Jul 14 2025

Keywords

Crossrefs

Cf. A000272, A156326, A385979, A385981, A385982.
Cf. A385946, 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, (j+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..3} binomial(3,k) * x^(k+1)/k! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385981 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * binomial(k+4,4) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 11, 526, 75981, 27017601, 20599793857, 30432196412318, 80590529100023889, 359767027014797719000, 2575966649397129017224661, 28392489655027195386265889544, 465411261102140455922541427819489, 11017701081052339904298545720453122836, 367264434033142995461894471693185212854475
Offset: 0

Views

Author

Seiichi Manyama, Jul 14 2025

Keywords

Crossrefs

Cf. A000272, A156326, A385979, A385980, A385982.

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, (j+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..4} binomial(4,k) * x^(k+1)/k! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385982 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * binomial(k+5,5) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 13, 856, 195525, 124248221, 188647130983, 611439299390984, 3879035706651051809, 44966039381652540837592, 900671755790709615794856671, 29761825253146859538914816137428, 1560353636451919718380582807368070417, 125541398272463750591414559674298911706684
Offset: 0

Views

Author

Seiichi Manyama, Jul 14 2025

Keywords

Crossrefs

Cf. A000272, A156326, A385979, A385980, A385981.

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, (j+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..5} binomial(5,k) * x^(k+1)/k! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

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
Previous Showing 11-19 of 19 results.

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