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 10 results.

A064880 Row sums of triangle A064879.

Original entry on oeis.org

1, 2, 2, 4, 11, 50, 389, 4590, 73950, 1559648, 41768100, 1372257648, 53854228438, 2482184155472, 132554850677690, 8096323604940164, 559278345518698923, 43305732768029924394, 3731115433469990603265
Offset: 0

Views

Author

Wolfdieter Lang, Oct 12 2001

Keywords

Links

Formula

a(n)= sum(A064879(n, m), m=0..n).

A064340 Generalized Catalan numbers C(2,2; n).

Original entry on oeis.org

1, 1, 4, 28, 256, 2704, 31168, 380608, 4840960, 63458560, 851399680, 11635096576, 161396604928, 2266669453312, 32166082822144, 460531091685376, 6644185553305600, 96498260064403456, 1409750653282287616, 20702370737659052032, 305428492830594039808
Offset: 0

Views

Author

Wolfdieter Lang, Oct 12 2001

Keywords

Comments

See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.

Links

Crossrefs

Cf. A000108 (Catalan as C(1,1; n)), A064879, A067298.

Programs

  • PARI
    my(x='x+O('x^30)); Vec((1+(13-3*sqrt(1-16*x))*x/2)/(1+2*x)^2) \\ Jinyuan Wang, Apr 20 2025

Formula

a(n) = ((4^(n-1))/(n-1))*Sum_{m=0..n-2} (m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)/2^(m+1), n >= 2, a(0) = a(1) = 1.
G.f.: (1-3*x*c(4*x))/(1-2*x*c(4*x))^2 = c(4*x)*(3+c(4*x))/(1+c(4*x))^2 = (1+5*x+3*c(4*x)*(2*x)^2)/(1+2*x)^2 with c(x) = A(x) g.f. of Catalan numbers A000108.
(-n+1)*a(n) + 2*(7*n-20)*a(n-1) + 16*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Aug 09 2017

A064341 Generalized Catalan numbers C(3,3; n).

Original entry on oeis.org

1, 1, 6, 81, 1566, 36126, 921456, 25055001, 711951606, 20891575566, 628237506276, 19259213633226, 599654171202156, 18911332670183856, 602840023457208516, 19392890824608619401, 628769286622411762086
Offset: 0

Views

Author

Wolfdieter Lang, Oct 12 2001

Keywords

Comments

See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.

Links

Crossrefs

Cf. A064340.

Formula

a(n) = ((9^(n-1))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/3)^(m+1)), m=0..n-2), n >= 2, a(0) := 1=: a(1).
G.f.: (1-5*x*c(9*x))/(1-3*x*c(9*x))^2 = c(9*x)*(5+4*c(9*x))/(1+2*c(9*x))^2 = (5*c(9*x)*(3*x)^2+4*(1+4*x))/(2+3*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.
2*(-n+1)*a(n) +3*(23*n-60)*a(n-1) +54*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017

A064342 Generalized Catalan numbers C(4,4; n).

Original entry on oeis.org

1, 1, 8, 176, 5888, 238848, 10770432, 518909952, 26156466176, 1362414338048, 72751723839488, 3961437637574656, 219123329636761600, 12278352550322765824, 695492547259800748032, 39759203500044029263872
Offset: 0

Views

Author

Wolfdieter Lang, Oct 12 2001

Keywords

Comments

See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.

Links

Crossrefs

A064341.

Formula

a(n)= ((16^(n-1))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/4)^(m+1)), m=0..n-2), n >= 2, a(0) := 1=: a(1).
G.f.:(1-7*x*c(16*x))/(1-4*x*c(16*x))^2 = c(16*x)*(7+9*c(16*x))/(1+3*c(16*x))^2 = (7*c(16*x)*(4*x)^2+3*(3+11*x))/(3+4*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.
3*(-n+1)*a(n) +4*(47*n-120)*a(n-1) +128*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017

A064343 Generalized Catalan numbers C(5,5; n).

Original entry on oeis.org

1, 1, 10, 325, 16750, 1056250, 74237500, 5580578125, 439118593750, 35714849218750, 2978473867187500, 253316015488281250, 21887247402929687500, 1915840314586914062500, 169529844641289062500000
Offset: 0

Views

Author

Wolfdieter Lang, Oct 12 2001

Keywords

Comments

See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.

Links

Crossrefs

Cf. A000108, A064342.

Formula

a(n) = ((25^(n-1))/(n-1))*Sum_{m=0..n-2} (m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/5)^(m+1)), n >= 2, a(0) := 1 =: a(1).
G.f.: (1-9*x*c(25*x))/(1-5*x*c(25*x))^2 = c(25*x)*(9+16*c(25*x))/(1+4*c(25*x))^2 = (9*c(25*x)*(5*x)^2+8*(2+7*x))/(4+5*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.
4*(-n+1)*a(n) +5*(79*n-200)*a(n-1) +250*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017

A064344 Generalized Catalan numbers C(6,6; n).

Original entry on oeis.org

1, 1, 12, 540, 39744, 3598992, 363776832, 39348690624, 4456429954560, 521760612125952, 62642882007530496, 7670452375558388736, 954216689151845302272, 120261048050627578368000
Offset: 0

Views

Author

Wolfdieter Lang, Oct 12 2001

Keywords

Comments

See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al.and Liggett references.

Links

Crossrefs

A064343.

Formula

a(n)= ((6^(2*(n-1)))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/6)^(m+1)), m=0..n-2), n >= 2, a(0) := 1=: a(1).
G.f.:(1-11*x*c(36*x))/(1-6*x*c(36*x))^2 = c(36*x)*(11+25*c(36*x))/(1+5*c(36*x))^2 = (11*c(36*x)*(6*x)^2+5*(5+17*x))/(5+6*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.
5*(-n+1)*a(n) +6*(119*n-300)*a(n-1) +432*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017

A064345 Generalized Catalan numbers C(7,7; n).

Original entry on oeis.org

1, 1, 14, 833, 83006, 10213854, 1404124008, 206635997673, 31844571309110, 5073749573133710, 829012595472718580, 138151786440502006186, 23390450962161609522028, 4012173837912126230070832
Offset: 0

Views

Author

Wolfdieter Lang, Oct 12 2001

Keywords

Comments

See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.

Links

Crossrefs

A064344.

Formula

a(n)= ((7^(2*(n-1)))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/7)^(m+1)), m=0..n-2), n >= 2, a(0) := 1=: a(1).
G.f.:(1-13*x*c(49*x))/(1-7*x*c(49*x))^2 = c(49*x)*(13+36*c(49*x))/(1+6*c(49*x))^2 = (13*c(49*x)*(7*x)^2+12*(3+10*x))/(6+7*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.
6*(-n+1)*a(n) +7*(167*n-420)*a(n-1) +686*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017

A064346 Generalized Catalan numbers C(8,8; n).

Original entry on oeis.org

1, 1, 16, 1216, 157696, 25317376, 4543676416, 873117515776, 175715349692416, 36562356662173696, 7802094251017240576, 1698089607837490610176, 375493988522687218057216, 84121868091432283370684416
Offset: 0

Views

Author

Wolfdieter Lang, Oct 12 2001

Keywords

Comments

See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.

Links

Crossrefs

A064345.

Formula

a(n)= ((8^(2*(n-1)))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/8)^(m+1)), m=0..n-2), n >= 2, a(0) := 1=: a(1).
G.f.:(1-15*x*c(64*x))/(1-8*x*c(64*x))^2 = c(64*x)*(15+49*c(64*x))/(1+7*c(64*x))^2 = (15*c(64*x)*(8*x)^2+7*(7+23*x))/(7+8*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.
7*(-n+1)*a(n) +8*(223*n-560)*a(n-1) +1024*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017

A064347 Generalized Catalan numbers C(9,9; n).

Original entry on oeis.org

1, 1, 18, 1701, 278478, 56542698, 12838905972, 3121895416077, 795077021525526, 209364566760439038, 56540432581528153788, 15573764062988183490786, 4358381303784085630372620, 1235729432868053981694246324
Offset: 0

Views

Author

Wolfdieter Lang, Oct 12 2001

Keywords

Comments

See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.

Links

Crossrefs

A064346.

Formula

a(n)= ((9^(2*(n-1)))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/9)^(m+1)), m=0..n-2), n >= 2, a(0) := 1=: a(1).
G.f.:(1-17*x*c(81*x))/(1-9*x*c(81*x))^2 = c(81*x)*(17+64*c(81*x))/(1+8*c(81*x))^2 = (17*c(81*x)*(9*x)^2+16*(4+13*x))/(8+9*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.
8*(-n+1)*a(n) +9*(287*n-720)*a(n-1) +1458*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 09 2017

A064878 Generalized Catalan numbers C(10,10; n).

Original entry on oeis.org

1, 1, 20, 2300, 464000, 116250000, 32580600000, 9779307000000, 3074524280000000, 999451946900000000, 333207298730000000000, 113305219025110000000000, 39145823948711200000000000
Offset: 0

Views

Author

Wolfdieter Lang, Oct 12 2001

Keywords

Comments

See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.

Crossrefs

A064347.

Formula

a(n)= ((10^(2*(n-1)))/(n-1))*sum((m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)*((1/10)^(m+1)), m=0..n-2), n >= 2, a(0) := 1=: a(1).
G.f.:(1-19*x*c(100*x))/(1-10*x*c(100*x))^2 = c(100*x)*(19+81*c(100*x))/(1+9*c(100*x))^2 = (19*c(100*x)*(10*x)^2+9*(9+29*x))/(9+10*x)^2 with c(x)= A(x) g.f. of Catalan numbers A000108.
Showing 1-10 of 10 results.

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