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 18 results. Next

A385835 a(n) = 1 + Sum_{k=0..n-1} (1 + k^2) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 7, 51, 660, 13350, 390886, 15728919, 836469748, 56989647229, 4849599126797, 504709937298467, 63117270187248665, 9344222191368190761, 1616899887657388367640, 323430766605746093449465, 74074314477265886578774322, 19261037812212680097678843345, 5643873902659784713257894768422
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Links

Crossrefs

Cf. A321087, A385836, A385837, A385838, A385839.
Cf. A385758.

Programs

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

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*A(x) - x^2 * (d/dx A(x)) - x^3 * (d^2/dx^2 A(x)) ) ).

A376125 a(n) = 1 + Sum_{k=0..n-1} (2*k+1) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 2, 9, 67, 681, 8556, 126253, 2124340, 39991633, 831271006, 18893178381, 465972248083, 12394713108433, 353750057246236, 10784915257548041, 349874160411051511, 12036066260440602401, 437714593034154481686, 16780944423208533034861, 676482338975579658794689
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 11 2024

Keywords

Links

Crossrefs

Cf. A000699, A007317, A321087.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[(2 k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x] - 2 x^2 A'[x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 / ( (1 - x) * (1 - x * A(x) - 2 * x^2 * A'(x)) ).
a(n) ~ c * 2^n * n * n!, where c = 0.6018110636400677977754542011395053310779724922160159... - Vaclav Kotesovec, Sep 11 2024
a(n) = 1 + n * Sum_{k=0..n-1} a(k) * a(n-1-k). - Seiichi Manyama, Jul 15 2025

A385758 G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x*A(x) - x^3*A''(x))).

Original entry on oeis.org

1, 2, 5, 25, 241, 3850, 92699, 3159424, 145529893, 8737971990, 664337673319, 62461188300465, 7121505696653881, 968606284138975286, 154985833403909522361, 28828521246104115576631, 6169483384435711859804021, 1505386674395483103372685258, 415493606617772745031305469471
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A321087, A385759, A385760, A385761.
Cf. A385762.

Programs

  • Mathematica
    terms = 19; A[] = 0; Do[A[x] = 1/((1-x)*(1-x*A[x]-x^3*A''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, (1+sum(k=1, 2, stirling(2, k, 1)*j^k))*v[j+1]*v[i-j])); v;

Formula

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

A385759 G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x*A(x) - x^4*A'''(x))).

Original entry on oeis.org

1, 2, 5, 15, 141, 3932, 251717, 31216948, 6680698525, 2271470142438, 1153913665217481, 835435792656039975, 830424340158140342961, 1099482665756962845820704, 1891111018270919721409143729, 4137752010118540256190073466415, 11312615890237585633045672755792789
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A321087, A385758, A385760, A385761.

Programs

  • Mathematica
    terms = 17; A[] = 0; Do[A[x] = 1/((1-x)*(1-x*A[x]-x^4*A'''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, (1+sum(k=1, 3, stirling(3, k, 1)*j^k))*v[j+1]*v[i-j])); v;

Formula

a(n) = 1 + Sum_{k=0..n-1} (1 + 2*k - 3*k^2 + k^3) * a(k) * a(n-1-k).

A385760 G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x*A(x) - x^5*A''''(x))).

Original entry on oeis.org

1, 2, 5, 15, 51, 1412, 175067, 63725638, 53784616915, 90573359145678, 274256185472187231, 1383348290257488337035, 10961652126528967555229301, 130268275255842369871718355444, 2235924687457083597476492688851325, 53724798520519979444347750309693062183
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A321087, A385758, A385759, A385761.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = 1/((1-x)*(1-x*A[x]-x^5*A''''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, (1+sum(k=1, 4, stirling(4, k, 1)*j^k))*v[j+1]*v[i-j])); v;

Formula

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

A385761 G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x*A(x) - x^6*A'''''(x))).

Original entry on oeis.org

1, 2, 5, 15, 51, 188, 23291, 16862710, 42561503035, 286183563337662, 4328240254531111671, 130903298544350358627387, 7257802488822060515691899445, 689810579878520205782663179307100, 106537105206016369903910237449838232525, 25594900303804029125790200935921438169789415
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A321087, A385758, A385759, A385760.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = 1/((1-x)*(1-x*A[x]-x^6*A'''''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, (1+sum(k=1, 5, stirling(5, k, 1)*j^k))*v[j+1]*v[i-j])); v;

Formula

a(n) = 1 + Sum_{k=0..n-1} (1 + 24*k - 50*k^2 + 35*k^3 - 10*k^4 + k^5) * a(k) * a(n-1-k).

A385838 a(n) = 1 + Sum_{k=0..n-1} (1 + k^5) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 7, 247, 61006, 62715298, 196236522104, 1526720482525833, 25665699044532909262, 841116296816234980686001, 49670440804927429155777517363, 4967242766473223753247263215133503, 799999284003076533259467892632499306811, 199068621859048073152067295737349123675521467
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A321087, A385835, A385836, A385837, A385839.
Cf. A385761, A385843.

Programs

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

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*A(x) - x*Sum_{k=1..5} Stirling2(5,k) * x^k * (d^k/dx^k A(x)) ) ).

A385836 a(n) = 1 + Sum_{k=0..n-1} (1 + k^3) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 7, 79, 2446, 166618, 21508712, 4732995201, 1642479584974, 847546182102241, 621260202463120771, 623749689526374747439, 832709044623310548285995, 1442255257225526024262579955, 3174408056872712362090099214740, 8723280646832436679639469748539639
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A321087, A385835, A385837, A385838, A385839.
Cf. A385759.

Programs

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

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*A(x) - x*Sum_{k=1..3} Stirling2(3,k) * x^k * (d^k/dx^k A(x)) ) ).

A385837 a(n) = 1 + Sum_{k=0..n-1} (1 + k^4) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 7, 135, 11472, 2983290, 1876558882, 2439543938823, 5867113337771476, 24055177364999767957, 157922269330003687462469, 1579854504025376907525660119, 23136970006572094830720177877037, 479860765871358769352536441406761329, 13707222893156109310485886790873337444816
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A321087, A385835, A385836, A385838, A385839.
Cf. A385760.

Programs

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

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*A(x) - x*Sum_{k=1..4} Stirling2(4,k) * x^k * (d^k/dx^k A(x)) ) ).

A385839 a(n) = 1 + Sum_{k=0..n-1} (1 + k^6) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 7, 471, 345240, 1415486250, 22122636527386, 1032242227753172079, 121446394933841583123508, 31836929544298684420302348229, 16919577022277987344334514604394117, 16919644700745370569015746375165719379327, 29974250364360598877961318618919670090162246645
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A321087, A385835, A385836, A385837, A385838.
Cf. A385834.

Programs

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

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*A(x) - x*Sum_{k=1..6} Stirling2(6,k) * x^k * (d^k/dx^k A(x)) ) ).
Showing 1-10 of 18 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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