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 14 results. Next

A386505 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * k^2 * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 43, 1717, 146261, 22851301, 5923208845, 2370243182889, 1386889039102537, 1137386506152214441, 1263728857603292729441, 1850186029852575829090909, 3487711314718246830637945549, 8300937715895750334611432889933, 24529666348754849148034163067487381
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A143925, A386506, A386507, A386508, A386509.
Cf. A385939, A386443.

Programs

  • Mathematica
    A386505[0] = 1;
A386505[n_] := A386505[n] = If[n==0,
            1,
            A386505[n-1]+ Sum[(1+k)*k^2*Binomial[n-1,k]*A386505[k]*A386505[n-1-k] ,{k,0,n-1} ]
        ] ;
Do [ Print[A386505[n]],{n,0,20}] (* R. J. Mathar, Aug 02 2025 *)
  • 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^2*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( x + x^2 * (d/dx A(x)) + x^3 * (d^2/dx^2 A(x)) ).

A386506 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * k^3 * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 79, 8845, 2875301, 2173904341, 3302241027205, 9087841330660905, 41958697476222137161, 306298931820000949752841, 3372659958223293180648888761, 53908617652925799897200239787869, 1211704268213547361986251511514073293, 37286568732242131447316119558759880633085
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A143925, A386505, A386507, A386508, A386509.
Cf. A385940, A386444.

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^3*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( x + x*Sum_{k=1..3} Stirling2(3,k) * x^k * (d^k/dx^k A(x)) ).

A386507 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * k^4 * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 151, 49525, 63641021, 239036610181, 2170214958201445, 41702857906969051017, 1537709560908537888618409, 100904503302575334820438217641, 11100605398391683050596962755215561, 1950420777626865443224119613333235611309, 525796384523344023260217345195483215249534941
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A143925, A386505, A386506, A386508, A386509.
Cf. A385941, A386445.

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^4*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( x + x*Sum_{k=1..4} Stirling2(4,k) * x^k * (d^k/dx^k A(x)) ).

A386508 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * k^5 * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 295, 287917, 1475577461, 27675935977381, 1506650312499716245, 202590228421127415254121, 59748112811137686928254493705, 35281260624146463343889980853779081, 38809774783723742261321649306513968984201, 75004702183951627532765950774478944180316824189
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A143925, A386505, A386506, A386507, A386509.
Cf. A385942, A386446.

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

Formula

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

A386509 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * k^6 * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 583, 1702357, 34872788861, 3269533221246901, 1067826281292319819285, 1005038096045094314876257929, 2371191405228277266497568590592937, 12601507027818562471139233302156639660841, 138616715922712004054565802733773706346507326441
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A143925, A386505, A386506, A386507, A386508.
Cf. A385943, A386447.

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

Formula

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

A385066 E.g.f. A(x) satisfies A(x) = exp(x + x^3*A''(x)).

Original entry on oeis.org

1, 1, 1, 7, 193, 12481, 1570201, 340513321, 117098181313, 60060238918849, 43839052690362481, 43879747204367814961, 58445034533293136385361, 101048138430710700967252945, 222098609829790469135187472009, 609650270727758340550662998605801, 2058153076335502227178904191401488641
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A143925, A386502, A386503, A386504.
Cf. A385920, A386448.

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)*sum(k=1, 2, stirling(2, k, 1)*j^k)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

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

A386502 E.g.f. A(x) satisfies A(x) = exp(x + x^4*A'''(x)).

Original entry on oeis.org

1, 1, 1, 1, 25, 3121, 1141921, 967142401, 1632504592321, 4951351715986369, 25004252825639317441, 198308457113999900437441, 2358282522829655305887600961, 40498770303734530275747011026561, 973509226030256545543333641850364737, 31906760631850274511535853878168004240641
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A143925, A385066, A386503, A386504.
Cf. A385921, A386449.

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)*sum(k=1, 3, stirling(3, k, 1)*j^k)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * (2*k - 3*k^2 + k^3) * binomial(n-1,k) * a(k) * a(n-1-k).
a(n) == 1 (mod 24). - Hugo Pfoertner, Jul 24 2025

A386510 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * binomial(k+1,2) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 34, 949, 52421, 5050711, 779516095, 181069531665, 60337677803905, 27766510630927741, 17108421087708824831, 13757393965653865220629, 14130398908817131991819653, 18201370833558663815315691987, 28941823262680770630349968403381, 56033750665620660972762531436196641
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A143925, A386511, A386512, A386513.
Cf. A386452, A386505.

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)*binomial(j+1, 2)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( x + x^2 * (d/dx A(x)) + x^3/2 * (d^2/dx^2 A(x)) ).

A386511 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * binomial(k+2,3) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 43, 1889, 198661, 42947941, 17142237365, 11658352652969, 12696712215226345, 21077148910182673081, 51239319321668728761281, 176469705716413028667777349, 837352955330191136544190873989, 5345677943448502627987168885274813, 44983970430636919384496638254796550221
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A143925, A386510, A386512, A386513.
Cf. A385945, A386453.

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)*binomial(j+2, 3)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( x + x*Sum_{k=1..3} binomial(2,k-1) * x^k/k! * (d^k/dx^k A(x)) ).

A386512 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * binomial(k+3,4) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 52, 3325, 598906, 255199051, 226888865575, 382997189880593, 1140957869006770561, 5659169551911928576531, 44571684957086887771692731, 535930324156886354251195391269, 9517054240482595566592327616630965, 242627830243798770154326313268171970697
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A143925, A386510, A386511, A386513.
Cf. A385946, A386454.

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)*binomial(j+3, 4)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( x + x*Sum_{k=1..4} binomial(3,k-1) * x^k/k! * (d^k/dx^k A(x)) ).
Showing 1-10 of 14 results. Next

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

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