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 1-6 of 6 results.

A364475 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x)^3.

Original entry on oeis.org

1, 1, 4, 18, 94, 529, 3135, 19270, 121732, 785496, 5155167, 34304706, 230923653, 1569684910, 10759159000, 74281473504, 516089542684, 3605685460750, 25316226436086, 178538289189108, 1264131169628799, 8982889404251721, 64041351551534215
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2023

Keywords

Crossrefs

Column k=1 of A378323.
Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364478.

Programs

  • Maple
    A364475 := proc(n)
    add( binomial(3*n-3*k,k) * binomial(3*n-4*k,n-2*k)/(2*n-2*k+1),k=0..n/2) ;
end proc:
seq(A364475(n),n=0..80); # R. J. Mathar, Jul 27 2023
  • PARI
    a(n) = sum(k=0, n\2, binomial(3*n-3*k, k)*binomial(3*n-4*k, n-2*k)/(2*n-2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-3*k,k) * binomial(3*n-4*k,n-2*k) / (2*n-2*k+1).
D-finite with recurrence 2*n*(2*n+1)*a(n) -(5*n+1)*(3*n-2)*a(n-1) +4*(-25*n^2+75*n-59) *a(n-2) +9*(-15*n^2+69*n-80)*a(n-3) -6*(3*n-8)*(3*n-10) *a(n-4)=0. - R. J. Mathar, Jul 27 2023

A104978 Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.

Original entry on oeis.org

1, 1, 1, 2, 5, 3, 5, 21, 28, 12, 14, 84, 180, 165, 55, 42, 330, 990, 1430, 1001, 273, 132, 1287, 5005, 10010, 10920, 6188, 1428, 429, 5005, 24024, 61880, 92820, 81396, 38760, 7752, 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263, 4862, 75582, 503880, 1899240, 4476780, 6864396, 6864396, 4326300, 1562275, 246675
Offset: 0

Views

Author

Paul D. Hanna, Mar 30 2005

Keywords

Examples

			The triangle T(n, k) begins:
  [0]    1;
  [1]    1,     1;
  [2]    2,     5,      3;
  [3]    5,    21,     28,     12;
  [4]   14,    84,    180,    165,     55;
  [5]   42,   330,    990,   1430,   1001,    273;
  [6]  132,  1287,   5005,  10010,  10920,   6188,   1428;
  [7]  429,  5005,  24024,  61880,  92820,  81396,  38760,   7752;
  [8] 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263;
  ...
The array A(n, k) begins:
  [0]   1,    1,      3,      12,       55,       273,       1428, ...  [A001764]
  [1]   1,    5,     28,     165,     1001,      6188,      38760, ...  [A025174]
  [2]   2,   21,    180,    1430,    10920,     81396,     596904, ...  [A383450]
  [3]   5,   84,    990,   10010,    92820,    813960,    6864396, ...  [A383451]
  [4]  14,  330,   5005,   61880,   678300,   6864396,   65615550, ...
  [5]  42, 1287,  24024,  352716,  4476780,  51482970,  551170620, ...
  [6] 132, 5005, 111384, 1899240, 27457584, 354323970, 4206302100, ...
  [A000108]  |  [A074922][A383452]
         [A002054]

Links

Crossrefs

Columns of array: A000108, A002054, A074922, A383452.
Rows of array: A001764, A025174, A383450, A383451.
Cf. A001002 (antidiagonal sums), A001764 (semidiagonal sums), A027307 (row sums), A104979, A383439 (central terms).

Programs

  • Magma
    [Binomial(2*n+k, n+2*k)*Binomial(n+2*k, k)/(n+k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 08 2021
  • Maple
    From Peter Luschny, May 04 2025:  (Start)
T := (n, k) -> (k + 2*n)!/(k!*(n - k)!*(n + k + 1)!):
seq(print(seq(T(n, k), k = 0..n)), n = 0..10);
# Alternatively the array:
A := (n, k) -> (3*k + 2*n)!/(k!*n!*(n + 2*k + 1)!);
for n from 0 to 8 do seq(A(n, k), k = 0..7) od;  (End)
  • Mathematica
    T[n_, k_]:= Binomial[2n+k, n+2k]*Binomial[n+2k, k]/(n+k+1);
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jan 27 2019 *)
  • PARI
    T(n,k) = local(A=1+x+x*y+x*O(x^n)+y*O(y^k)); for(i=1,n,A=1+x*A^2+x*y*A^3); polcoeff(polcoeff(A,n,x),k,y)
for(n=0, 10, for(k=0, n, print1(T(n,k),", ")); print(""))
  • PARI
    Dy(n, F)=local(D=F); for(i=1, n, D=deriv(D,y)); D
T(n,k)=local(A=1); A=1+sum(m=1, n+1, x^m/y^(m+1) * Dy(m-1, (y^2+y^3)^m/m!)) +x*O(x^n)+y*O(y^k); polcoeff(polcoeff(A, n,x),k,y)
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Jun 22 2012
  • PARI
    x='x; y='y; z='z; Fxyz = 1 - z + x*z^2 + x*y*z^3;
seq(N) = {
  my(z0 = 1 + O((x*y)^N), z1 = 0);
  for (k = 1, N^2,
    z1 = z0 - subst(Fxyz, z, z0)/subst(deriv(Fxyz, z), z, z0);
    if (z0 == z1, break()); z0 = z1);
  vector(N, n, Vecrev(polcoeff(z0, n-1, 'x)));
};
concat(seq(9)) \\ Gheorghe Coserea, Nov 30 2016
  • Sage
    flatten([[binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 08 2021

Formula

T(n, k) = binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1).
G.f.: A(x, y) = Sum_{n>=0} x^n/y^(n+1) * d^(n-1)/dy^(n-1) (y^2 + y^3)^n / n!. - Paul D. Hanna, Jun 22 2012
G.f. of row n: 1/y^(n+1) * d^(n-1)/dy^(n-1) (y^2+y^3)^n / n!. - Paul D. Hanna, Jun 22 2012
A(n, k) = T(n + k, k) = (3*k + 2*n)! / (k!*n!*(n + 2*k + 1)!). - Peter Luschny, May 04 2025

A364474 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x).

Original entry on oeis.org

1, 1, 4, 16, 77, 403, 2228, 12800, 75653, 457022, 2809266, 17514200, 110480475, 703850686, 4522217364, 29268545416, 190645760149, 1248817411471, 8221323983431, 54365667330636, 360954069730636, 2405225494066647, 16080210766344354, 107828663888705292
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2023

Keywords

Links

Crossrefs

Cf. A002293, A104979, A186997, A255673, A361245, A364475, A364478.

Programs

  • Maple
    A364474 := proc(n)
    add( binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k)/(2*n-4*k+1),k=0..n/2) ;
end proc:
seq(A364474(n),n=0..80); # R. J. Mathar, Jul 27 2023
  • Mathematica
    Table[Sum[Binomial[3*n - 5*k, k]*Binomial[3*n - 6*k, n - 2*k]/(2*n - 4*k + 1), {k, 0, Floor[n/2]}], {n, 0, 25}] (* Wesley Ivan Hurt, May 25 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, binomial(3*n-5*k, k)*binomial(3*n-6*k, n-2*k)/(2*n-4*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k) / (2*n-4*k+1).
D-finite with recurrence 2*n*(2*n+1)*(3*n-7)*a(n) -3*(3*n-1)*(3*n-7)*(3*n-2) *a(n-1) -2*(n-3)*(18*n^2-33*n+4) *a(n-2) +2*(18*n^3-141*n^2+287*n-64) *a(n-4) -2*(n-4)*(3*n-1)*(2*n-13)*a(n-6)=0. - R. J. Mathar, Jul 27 2023

A200755 G.f. satisfies A(x) = 1 + x*A(x)^3 - x^2*A(x)^2.

Original entry on oeis.org

1, 1, 2, 7, 29, 129, 602, 2910, 14447, 73234, 377487, 1972568, 10425930, 55640282, 299403552, 1622701202, 8850030065, 48534971244, 267486182192, 1480673755443, 8228819436898, 45895682480965, 256815165790211, 1441321638029496, 8111194646903282
Offset: 0

Views

Author

Paul D. Hanna, Nov 21 2011

Keywords

Comments

Compare to the g.f. C(x) for the Catalan numbers (A000108): C(x) = 1 + x*C(x)^3 - x^2*C(x)^4 = 1 + x*C(x)^2.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 129*x^5 + 602*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 76*x^4 + 344*x^5 + 1627*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 147*x^4 + 678*x^5 + 3254*x^6 +...
where a(2) = 3 - 1; a(3) = 9 - 2; a(4) = 34 - 5; a(5) = 147 - 18; ...

Crossrefs

Cf. A104979, A200754.
Cf. A000108, A137265, A200753.

Programs

  • Mathematica
    nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[1 + x*AGF^3 - x^2*AGF^2 - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 10 2013 *)
  • PARI
    a(n)=local(A=1+x);for(i=1,n,A=1+x*A^3-x^2*A^2+x*O(x^n));polcoeff(A,n);
  • PARI
    a(n) = sum(k=0, n\2, (-1)^k*binomial(3*n-4*k, k)*binomial(3*n-5*k, n-2*k)/(2*n-3*k+1)); \\ Seiichi Manyama, Nov 02 2023

Formula

Recurrence: 2*n*(2*n+1)*(244*n^3 - 1713*n^2 + 3767*n - 2550)*a(n) = 3*(2196*n^5 - 17613*n^4 + 49628*n^3 - 59841*n^2 + 30478*n - 5184)*a(n-1) - 18*(244*n^5 - 2323*n^4 + 8013*n^3 - 12252*n^2 + 7774*n - 1260)*a(n-2) - (n-4)*(244*n^4 - 1713*n^3 + 4172*n^2 - 3333*n - 378)*a(n-3) - 36*(n-5)*(n-3)*(5*n + 2)*a(n-4) - 4*(n-6)*(n-4)*(244*n^3 - 981*n^2 + 1073*n - 252)*a(n-5). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = 5.991151107674316485... is the root of the equation -4 - 4*d - 5*d^2 - 23*d^3 + 4*d^4 = 0 and c = 0.214566307956522153666714736272121899143... - Vaclav Kotesovec, Sep 10 2013
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-4*k,k) * binomial(3*n-5*k,n-2*k) / (2*n-3*k+1). - Seiichi Manyama, Nov 02 2023

A364478 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x)^8.

Original entry on oeis.org

1, 1, 4, 23, 154, 1124, 8675, 69626, 575243, 4859778, 41789764, 364565277, 3218581695, 28702642553, 258172627259, 2339496034381, 21337716782873, 195726876816623, 1804472496834650, 16711389876481027, 155395461519245354, 1450298253483719944
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2023

Keywords

Crossrefs

Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(3*n+2*k, k)*binomial(3*n+k, n-2*k)/(2*n+3*k+1));

Formula

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

A367040 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^3.

Original entry on oeis.org

1, 1, 4, 15, 70, 360, 1953, 11008, 63837, 378390, 2282205, 13960890, 86411232, 540166219, 3405341160, 21625820793, 138216775785, 888371346825, 5738510504979, 37234351046835, 242567430368298, 1585979835198675, 10403866383915844, 68453912880893025
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2023

Keywords

Crossrefs

Cf. A176697, A367041.
Cf. A366266, A366676.
Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475, A364478.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(2*(n-2*k)+1, k)*binomial(3*(n-2*k), n-2*k)/(2*(n-2*k)+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(2*(n-2*k)+1,k) * binomial(3*(n-2*k),n-2*k)/(2*(n-2*k)+1).
Showing 1-6 of 6 results.

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

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