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 1-8 of 8 results.

A385830 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k^2) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 20, 241, 4623, 130300, 5100750, 265780029, 17827454651, 1498498011875, 154408489507578, 19151761451917580, 2815820822235814540, 484383420815495253624, 96401320782466194458886, 21981036279413999807199045, 5693391431445001330242504699, 1662538953499888924638316487305
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Links

Crossrefs

Cf. A088716, A385831, A385832, A385833, A385834.
Cf. A385762, A385835.

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^2)*v[j+1]*v[i-j])); v;

Formula

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

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).

A385763 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x) - x^4*A'''(x)).

Original entry on oeis.org

1, 1, 2, 5, 44, 1188, 74880, 9211479, 1962123260, 665169218468, 337242780292376, 243827199998597254, 242120748323922920272, 320325994582940359050400, 550640627320172764415124000, 1204251372776149567847238889047, 3291219553094816112273747054673476
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A000108, A088716, A385762, A385764, A385765.

Programs

  • Mathematica
    terms = 17; A[] = 0; Do[A[x] = 1/(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)); v[1]=1; for(i=1, n, v[i+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(0) = 1; a(n) = Sum_{k=0..n-1} (1 + 2*k - 3*k^2 + k^3) * a(k) * a(n-1-k).

A385764 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x) - x^5*A''''(x)).

Original entry on oeis.org

1, 1, 2, 5, 14, 378, 46500, 16879869, 14229776750, 23948731244678, 72492823741526156, 365581334105823084634, 2896500982661242290253612, 34419121542689992919239814260, 590735385934420874267059790772360, 14193599152271246770955912922939691797, 465024889616667096875210999651863472880846
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A000108, A088716, A385762, A385763, A385765.

Programs

  • Mathematica
    terms = 17; A[] = 0; Do[A[x] = 1/(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)); v[1]=1; for(i=1, n, v[i+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(0) = 1; a(n) = Sum_{k=0..n-1} (1 - 6*k + 11*k^2 - 6*k^3 + k^4) * a(k) * a(n-1-k).

A385765 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x) - x^6*A'''''(x)).

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 5172, 3739389, 9434483630, 63428037194102, 959222215928392076, 29009757539769286481866, 1608387988236777669667251772, 152866019594999736359695792369300, 23609086665918990295149462904374925800, 5671917808033245221993631555503554148332485
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A000108, A088716, A385762, A385763, A385764.
Cf. A385761.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = 1/(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)); v[1]=1; for(i=1, n, v[i+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(0) = 1; a(n) = 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).

A385920 E.g.f. A(x) satisfies A(x) = exp(x*A(x) + x^3*A''(x)).

Original entry on oeis.org

1, 1, 3, 34, 1085, 76176, 10075567, 2259237184, 795650626521, 415436957516800, 307467426910853051, 311183690415601457664, 418253671031607891057877, 728624453608629352377831424, 1611758187912750506708147828775, 4448533739124778044473142239512576
Offset: 0

Views

Author

Seiichi Manyama, Jul 12 2025

Keywords

Crossrefs

Cf. A000272, A156326, A385921, A385922, A385923.
Cf. A385762.

Programs

  • Mathematica
    terms = 16; A[] = 1; Do[A[x] = Exp[x*A[x]+x^3*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, 2, stirling(2, 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^3) * binomial(n-1,k) * a(k) * a(n-1-k).

A386448 G.f. A(x) satisfies A(x) = 1/(1 - x - x^3*A''(x)).

Original entry on oeis.org

1, 1, 1, 3, 23, 319, 6999, 223725, 9838405, 570440733, 42203958765, 3882243620535, 434771830226307, 58255737747374083, 9203989127308306571, 1693477639607917108953, 359008305377998952818761, 86878355403079952880852217, 23804317478591173659253678809, 7331644401028481860472940727371
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2025

Keywords

Crossrefs

Cf. A075834, A386449, A386450, A386451.
Cf. A385762.

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

Formula

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

A385844 G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x^3*A''(x))).

Original entry on oeis.org

1, 1, 1, 3, 21, 273, 5737, 177919, 7651849, 436186313, 31842549569, 2897710853939, 321648004495773, 42779331295225353, 6716367934603667145, 1229096733282700520799, 259339594018913458094865, 62500870590534491566841265, 17062742827503910747790541249, 5238263128497776755775631825219
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A143917, A385845, A385846.
Cf. A385758, A385762.

Programs

  • Mathematica
    terms = 20; A[] = 0; Do[A[x] = 1/((1 - x) * (1 - x^3*A''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]  (* Stefano Spezia, Jul 10 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-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} (-k + k^2) * a(k) * a(n-1-k).
Showing 1-8 of 8 results.

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

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