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.

A116880 Generalized Catalan triangle, called CM(1,2).

Original entry on oeis.org

1, 1, 3, 3, 7, 13, 13, 29, 41, 67, 67, 147, 195, 247, 381, 381, 829, 1069, 1277, 1545, 2307, 2307, 4995, 6339, 7379, 8451, 9975, 14589, 14589, 31485, 39549, 45373, 50733, 56829, 66057, 95235, 95235, 205059, 255747, 290691, 320707, 351187, 388099, 446455, 636925
Offset: 0

Views

Author

Wolfdieter Lang, Mar 24 2006

Keywords

Comments

This triangle generalizes the 'new' Catalan triangle A028364 (which could be called CM(1,1); M stands for author Meeussen).

Examples

			Triangle begins:
     1;
     1,    3;
     3,    7,   13;
    13,   29,   41,   67;
    67,  147,  195,  247,  381;
   381,  829, 1069, 1277, 1545, 2307;
  2307, 4995, 6339, 7379, 8451, 9975, 14589;

Links

Crossrefs

Column m=0 gives A064062.
Row sums give A116881.

Programs

  • Maple
    lim:=8: c:=(1-sqrt(1-8*x))/(4*x): g:=(1+2*x*c)/(1+x): gf1:=g*(x*c)^m: for m from 0 to lim do t:=taylor(gf1, x, lim+1): for n from 0 to lim do a[n,m]:=coeff(t, x, n):od:od: gf2:=g*sum(a[s,k]*(2*c)^k,k=0..s): for s from 0 to lim do t:=taylor(gf2, x, lim+1): for n from 0 to lim do b[n,s]:=coeff(t, x, n):od:od: seq(seq(b[n-s,s],s=0..n),n=0..lim); # Nathaniel Johnston, Apr 30 2011

Formula

G.f. for columns m >= 0 (without leading zeros): c(2;x)*Sum_{k=0..m} C(1,2;m,k)*(2*c(2*x))^k with c(2;x):=(1+2*x*c(2*x))/(1+x) the g.f. of A064062 and c(x) is the g.f. of A000108 (Catalan). C(1,2;n,m) is the triangle A115193(n,m).

A386834 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 8, 111, 1738, 28701, 488412, 8473387, 148994510, 2645999673, 47349481408, 852429930567, 15421507805106, 280126256513109, 5105764838932388, 93331970924544099, 1710369544783134614, 31412304686874624113, 578023658034894471048, 10654486069487503147135
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A116881, A386833.
Cf. A370101, A386837.
Cf. A383716.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(4*n-k-1, n-k));

Formula

a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(3*n).
a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^(3*n)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(3*n+k-1,k).
Conjecture D-finite with recurrence +3*n*(16543753*n -26995933)*(3*n-1)*(3*n-2)*a(n) +(-1669899251*n^4 -26931977989*n^3 +131963667975*n^2 -188283072995*n +85757456660)*a(n-1) +2*(-61301926003*n^4 +515926265010*n^3 -1655392333929*n^2 +2311146075302*n -1165379619540)*a(n-2) -96*(39221117*n -50949760)*(4*n-9)*(2*n-5)*(4*n-7)*a(n-3)=0. - R. J. Mathar, Aug 19 2025

A386833 a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(3*n-k-1,n-k).

Original entry on oeis.org

1, 6, 59, 656, 7701, 93210, 1150495, 14395428, 181936169, 2317140014, 29691138099, 382334271544, 4943464235069, 64137141682242, 834561532624967, 10886878474010700, 142332442919829585, 1864423992564121686, 24464149489904517211, 321499324010641490016, 4230840338116037836901
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A116881, A386834.
Cf. A370097, A386836.
Cf. A383326.

Programs

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

Formula

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

A386835 a(n) = Sum_{k=0..n} binomial(2*n+2,k) * binomial(2*n-k-1,n-k).

Original entry on oeis.org

1, 5, 30, 198, 1375, 9843, 71876, 532220, 3981645, 30023265, 227803642, 1737227682, 13303481035, 102234258623, 787997000640, 6089345072056, 47161769198809, 365986358229645, 2845097133606422, 22151577531840830, 172710278146819959, 1348274852150114251
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A386836, A386837.
Cf. A116881.

Programs

  • Mathematica
    Table[Sum[Binomial[2*n + 2, k]*Binomial[2*n - k - 1, n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(2*n+2, k)*binomial(2*n-k-1, n-k));

Formula

a(n) = [x^n] (1+x)^(2*n+2)/(1-x)^n.
a(n) = [x^n] 1/((1-x)^3 * (1-2*x)^n).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(2*n+2,k) * binomial(n-k+2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k-1,k) * binomial(n-k+2,n-k).
a(n) ~ 2^(3*n+5) / (27*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 06 2025

A386843 a(n) = Sum_{k=0..n} binomial(2*n+2,k) * binomial(2*n-k,n-k).

Original entry on oeis.org

1, 6, 39, 268, 1905, 13842, 102123, 761880, 5732325, 43417630, 330620895, 2528772132, 19412942809, 149497184298, 1154365194195, 8934458916912, 69291946278861, 538372925816886, 4189702003359687, 32651982699233340, 254800541773725633, 1990683254889381954
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A386844, A386845.
Cf. A059304, A178792.
Cf. A116881.

Programs

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

Formula

a(n) = [x^n] (1+x)^(2*n+2)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(2*n+2,k).
a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(n+k,k).

A386862 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).

Original entry on oeis.org

1, 5, 42, 409, 4238, 45414, 496996, 5517929, 61909878, 700189606, 7968994124, 91158632250, 1047156227068, 12071222381456, 139569181458552, 1617879480097129, 18796461329347238, 218806784598226926, 2551538498649588892, 29800118958422522414, 348529038403155280548
Offset: 0

Views

Author

Seiichi Manyama, Aug 06 2025

Keywords

Crossrefs

Cf. A383888, A386865.
Cf. A116881.

Programs

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

Formula

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

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