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.

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A158836 Column 0 of triangle A158835.

Original entry on oeis.org

1, 1, 4, 27, 254, 3062, 45052, 783151, 15712342, 357459042, 9094926988, 255939571048, 7893741230500, 264806871279676, 9600056691219936, 374033821840909263, 15586672520501193866, 691789220336675178652
Offset: 0

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Links

Crossrefs

Cf. A158835, A158837, A158838, A158839, A158840.

Programs

  • PARI
    {a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))),M,N,P);M=matrix(n+2, n+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+1, n+1, r, c, M[r, c]); P=matrix(n+1, n+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1,1]}

A158837 Column 1 of triangle A158835.

Original entry on oeis.org

1, 2, 11, 94, 1072, 15212, 257777, 5074738, 113775490, 2861365660, 79763482974, 2440866020252, 81343355108428, 2932370770780016, 113695507437209845, 4717853729131352186, 208615291319607614600, 9792578421235713418464
Offset: 0

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Crossrefs

Cf. A158835, A158836, A158838, A158839, A158840.

Programs

  • PARI
    {a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+3))),M,N,P);M=matrix(n+3, n+3, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+2, n+2, r, c, M[r, c]); P=matrix(n+2, n+2, r, c, M[r+1, c]); (P~*N~^-1)[n+2,2]}

A158839 Column 3 of triangle A158835.

Original entry on oeis.org

1, 4, 34, 412, 6325, 116372, 2483706, 60168736, 1628677692, 48672911296, 1590752204044, 56418074957840, 2157411204773415, 88464995576660084, 3871611011946560294, 180101399407072883012, 8873328068327122625596
Offset: 0

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Crossrefs

Cf. A158835, A158836, A158837, A158836, A158840.

Programs

  • PARI
    {a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+5))),M,N,P);M=matrix(n+5, n+5, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+4, n+4, r, c, M[r, c]); P=matrix(n+4, n+4, r, c, M[r+1, c]); (P~*N~^-1)[n+4,4]}

A158840 Row sums of triangle A158835.

Original entry on oeis.org

1, 2, 7, 42, 374, 4391, 63637, 1095362, 21823226, 493898216, 12515588806, 351062669154, 10798972965266, 361471373319171, 13080119556342713, 508813238759275712, 21174032937728251318, 938646693399848483498
Offset: 0

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Crossrefs

Cf. A158835, A158836, A158837, A158838, A158839.

Programs

  • PARI
    {a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))), M, N, P); M=matrix(n+2, n+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+1, n+1, r, c, M[r, c]); P=matrix(n+1, n+1, r, c, M[r+1, c]); sum(k=0,n,(P~*N~^-1)[n+1, k+1])}

A158827 The 4th iteration of x*C(x) where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 4, 20, 110, 640, 3870, 24084, 153306, 993978, 6544242, 43652340, 294469974, 2006018748, 13784115468, 95444016984, 665407010349, 4667570034444, 32922870719664, 233389493503968, 1662048903052380, 11885333877149532
Offset: 1

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Links

Crossrefs

Cf. A158825, A158826, A158828, A158829.

Programs

  • PARI
    {a(n)=local(F=serreverse(x-x^2+O(x^(n+1))),G=x); for(i=1,4,G=subst(F,x,G));polcoeff(G,n)}

Formula

Series reversion of 1 -4*x +12*x^2 -30*x^3 +64*x^4 -118*x^5 +188*x^6 -258*x^7 +302*x^8 -298*x^9 +244*x^10 -162*x^11 +84*x^12 -32*x^13 +8*x^14 -x^15. - R. J. Mathar, Aug 30 2021

A158828 The 5th iteration of x*C(x) where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 5, 30, 195, 1330, 9380, 67844, 500619, 3755156, 28558484, 219767968, 1708590960, 13403300208, 105983648060, 844009565176, 6764300053390, 54525119251104, 441811163402124, 3597005618194848, 29412560840221272
Offset: 1

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Links

Crossrefs

Cf. A158825, A158826, A158827, A158829.

Programs

  • PARI
    {a(n)=local(F=serreverse(x-x^2+O(x^(n+1))),G=x); for(i=1,5,G=subst(F,x,G));polcoeff(G,n)}

Formula

Series reversion of x -5*x^2 +20*x^3 -70*x^4 +220*x^5 -630*x^6 +1656*x^7 -4014*x^8 +8994*x^9 -18654*x^10 +35832*x^11 -63750*x^12 +105024*x^13 -160120*x^14 +225696*x^15 -293685*x^16 +352074*x^17 -387820*x^18 +391232*x^19 -359992*x^20 +300664*x^21 -226580*x^22 +152952*x^23 -91656*x^24 +48204*x^25 -21924*x^26 +8456*x^27 -2692*x^28 +680*x^29 -128*x^30 +16*x^31 -x^32. - R. J. Mathar, Aug 30 2021

A158838 Column 2 of triangle A158835.

Original entry on oeis.org

1, 3, 21, 217, 2904, 47337, 906557, 19910808, 492818850, 13564326950, 410807572044, 13573135469214, 485765085176420, 18717987193565613, 772565258231236269, 34002334709760133807, 1589555183231724515700
Offset: 0

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Crossrefs

Cf. A158835, A158836, A158837, A158839, A158840.

Programs

  • PARI
    {a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+4))),M,N,P);M=matrix(n+4, n+4, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+3, n+3, r, c, M[r, c]); P=matrix(n+3, n+3, r, c, M[r+1, c]); (P~*N~^-1)[n+3,3]}

A166906 Column 1 of triangle A166905.

Original entry on oeis.org

1, 4, 33, 380, 5510, 95732, 1933288, 44437080, 1144564278, 32638644236, 1020503373032, 34708182795156, 1275532011982176, 50365443858930384, 2126358227959866224, 95577781657788563192, 4556923094838105968302
Offset: 0

Views

Author

Paul D. Hanna, Nov 28 2009

Keywords

Comments

Triangle A166905 transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).

Crossrefs

Cf. A166905, A158831, A166907, A166908, A166909.

Programs

  • PARI
    {a(n)=local(F=x, G=serreverse(x-x^2+O(x^(n+4))), M, N, P, m=n); M=matrix(m+3, m+3, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+3)))); polcoeff(F, c)); N=matrix(m+2, m+2, r, c, F=x;for(i=1, r, F=subst(F, x, G+x*O(x^(m+3)))); polcoeff(F, c)); P=matrix(m+2, m+2, r, c, M[r+1, c]); (P~*N~^-1)[n+2, 2]}

A166907 Column 2 of triangle A166905.

Original entry on oeis.org

1, 9, 108, 1610, 28560, 586320, 13658904, 355787568, 10243342296, 322939137312, 11063339361360, 409194048521778, 16249995494795920, 689585033717023224, 31140529927119263136, 1490994828293677370148, 75444108490820383882392
Offset: 0

Views

Author

Paul D. Hanna, Nov 28 2009

Keywords

Comments

Triangle A166905 transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).

Crossrefs

Cf. A166905, A158831, A166906, A166908, A166909.

Programs

  • PARI
    {a(n)=local(F=x, G=serreverse(x-x^2+O(x^(n+5))), M, N, P, m=n); M=matrix(m+4, m+4, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+4)))); polcoeff(F, c)); N=matrix(m+3, m+3, r, c, F=x;for(i=1, r, F=subst(F, x, G+x*O(x^(m+4)))); polcoeff(F, c)); P=matrix(m+3, m+3, r, c, M[r+1, c]); (P~*N~^-1)[n+3, 3]}

A166908 Column 3 of triangle A166905.

Original entry on oeis.org

1, 16, 270, 5148, 110929, 2677008, 71648322, 2107966432, 67649839664, 2352412120760, 88122951915388, 3538364803586104, 151611580761978784, 6905283671128114400, 333151832685664811338, 16973306740660778801468
Offset: 0

Views

Author

Paul D. Hanna, Nov 28 2009

Keywords

Comments

Triangle A166905 transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).

Crossrefs

Cf. A166905, A158831, A166906, A166907, A166909.

Programs

  • PARI
    {a(n)=local(F=x, G=serreverse(x-x^2+O(x^(n+6))), M, N, P, m=n); M=matrix(m+5, m+5, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+5)))); polcoeff(F, c)); N=matrix(m+4, m+4, r, c, F=x;for(i=1, r, F=subst(F, x, G+x*O(x^(m+5)))); polcoeff(F, c)); P=matrix(m+4, m+4, r, c, M[r+1, c]); (P~*N~^-1)[n+4, 4]}
Previous Showing 11-20 of 23 results. Next

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