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 1-3 of 3 results.

A137572 The first upper diagonal of square array A137570; equals the convolution of the main diagonal A137571 with A002293.

Original entry on oeis.org

1, 3, 16, 100, 681, 4908, 36842, 285158, 2260257, 18257902, 149769225, 1244277499, 10448404901, 88538107802, 756153001241, 6501989278168, 56244305146039, 489111092027854, 4273491476147117, 37496699100314116, 330261353255659842
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2008

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 16*x^2 + 100*x^3 + 681*x^4 + 4908*x^5 +...;
A(x) = F(x)/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2n,n)/(n+1), ...] and
F(x) = 1 + xF(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4n,n)/(3n+1), ...].

Crossrefs

Cf. A137570, A137571, A137573; A000108, A002293.

Programs

  • PARI
    {a(n)=local(m=n+1,C,F,A); C=Ser(vector(m,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(m,r,binomial(4*r-4,r-1)/(3*r-2))); A=F/(1-x*C*F^2-x*F^3);polcoeff(A+O(x^m),n,x)}

Formula

G.f. A(x) = F(x)/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108) and F(x) = 1 + xF(x)^4 is g.f. of A002293.

A137573 The first lower diagonal in square array A137570; equals the convolution of the main diagonal A137571 with the Catalan numbers (A000108) and with the square of A002293.

Original entry on oeis.org

1, 5, 29, 186, 1281, 9294, 70109, 544833, 4333381, 35108351, 288738813, 2404256945, 20228988678, 171716799066, 1468804301441, 12647321103329, 109538312419238, 953622158606749, 8340394595266367, 73247287493299642
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2008

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 29*x^2 + 186*x^3 + 1281*x^4 + 9294*x^5 +...;
A(x) = C(x)*F(x)^2/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2n,n)/(n+1), ...] and
F(x) = 1 + xF(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4n,n)/(3n+1), ...].

Crossrefs

Cf. A137570, A137571, A137572; A000108, A002293.

Programs

  • PARI
    {a(n)=local(m=n+1,C,F,A); C=Ser(vector(m,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(m,r,binomial(4*r-4,r-1)/(3*r-2))); A=C*F^2/(1-x*C*F^2-x*F^3);polcoeff(A+O(x^m),n,x)}

Formula

G.f. A(x) = C(x)*F(x)^2/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108) and F(x) = 1 + xF(x)^4 is g.f. of A002293.

A137570 Square array, read by antidiagonals, where row n+1 equals the partial sums of the previous row after removing the terms in positions {n, n+1} from row n for n>=0, with row 0 equal to all 1's.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 6, 10, 4, 1, 1, 7, 29, 16, 5, 1, 1, 8, 36, 60, 23, 6, 1, 1, 9, 44, 186, 100, 31, 7, 1, 1, 10, 53, 230, 397, 150, 40, 8, 1, 1, 11, 63, 283, 1281, 681, 211, 50, 9, 1, 1, 12, 74, 346, 1564, 2802, 1051, 284, 61, 10, 1, 1, 13, 86, 420, 1910, 9294, 4908
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2008

Keywords

Examples

			Square array begins:
  (1),(1), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
  1,(2),(3), 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...;
  1, 5,(10),(16), 23, 31, 40, 50, 61, 73, 86, 100, 115, 131, 148, ...;
  1, 6, 29,(60),(100), 150, 211, 284, 370, 470, 585, 716, 864, ...;
  1, 7, 36, 186,(397),(681), 1051, 1521, 2106, 2822, 3686, 4716, ...;
  1, 8, 44, 230, 1281,(2802),(4908), 7730, 11416, 16132, 22063, ...;
  1, 9, 53, 283, 1564, 9294,(20710),(36842), 58905, 88319, 126730, ...;
  1, 10, 63, 346, 1910, 11204, 70109,(158428),(285158), 461190, ...;
  1, 11, 74, 420, 2330, 13534, 83643, 544833,(1244413),(2260257), ...;
  ...
For each row, remove the terms along the diagonals (in parenthesis),
and then take partial sums to obtain the next row.
GENERATING FUNCTIONS.
The g.f. of n-th lower diagonal equals D(x)*F(x)^2*C(x)^n and
the g.f. of n-th upper diagonal equals D(x)*F(x)^n,
where D(x) is g.f. of main diagonal (A137571):
[1, 2, 10, 60, 397, 2802, 20710, 158428, 1244413, 9980220, ...]
defined by:
D(x) = 1/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + x*C(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2*n,n)/(n+1), ...] and
F(x) = 1 + x*F(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4*n,n)/(3*n+1), ...].

Crossrefs

Cf. A130523 (variant); diagonals: A137571, A137572, A137573; related: A000108, A002293.

Programs

  • PARI
    T(n, k)=if(k<0, 0, if(n==0, 1, T(n, k-1) + if(n-1>k, T(n-1, k), T(n-1, k+2))))
  • PARI
    /* Using Formula for G.F.: */ T(n,k)=local(m=max(n,k)+1,C,F,D,A); C=subst(Ser(vector(m,r,binomial(2*r-2,r-1)/r)),x,x*y); F=subst(Ser(vector(m,r,binomial(4*r-4,r-1)/(3*r-2))),x,x*y); D=1/(1-x*y*C*F^2-x*y*F^3); A=D*(1/(1-y*F) + x*C*F^2/(1-x*C)); polcoeff(polcoeff(A+O(x^m),n,x)+O(y^m),k,y)

Formula

G.f.: A(x,y) = D(x*y)*(1/(1 - y*F(x*y)) + x*C(x*y)*F(x*y)^2/(1 - x*C(x*y))), where D(x) = 1/(1 - x*C(x)*F(x)^2 - x*F(x)^3) is the g.f. of the main diagonal (A137571), C(x) = g.f. of Catalan numbers (A000108) and F(x) = g.f. of A002293; thus the g.f. of n-th lower diagonal = D(x)*F(x)^2*C(x)^n and the g.f. of n-th upper diagonal = D(x)*F(x)^n.
Showing 1-3 of 3 results.

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