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 1-4 of 4 results.

A130523 Square array, read by antidiagonals, where row n+1 equals the partial sums of the previous row after removing the n-th term 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, 4, 3, 1, 1, 5, 8, 4, 1, 1, 6, 18, 13, 5, 1, 1, 7, 24, 37, 19, 6, 1, 1, 8, 31, 87, 63, 26, 7, 1, 1, 9, 39, 118, 184, 97, 34, 8, 1, 1, 10, 48, 157, 442, 324, 140, 43, 9, 1, 1, 11, 58, 205, 599, 959, 517, 193, 53, 10, 1, 1, 12, 69, 263, 804, 2332, 1733, 774, 257, 64, 11
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2007, Jun 06 2007

Keywords

Comments

The g.f. of n-th lower diagonal equals D(x)*F(x)*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 (A007857), C(x) is g.f. of Catalan numbers (A000108) and F(x) is g.f. of ternary numbers (A001764).

Examples

			Square array begins:
  (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, 4, (8), 13, 19, 26, 34, 43, 53, 64, 76, 89, 103, 118, 134, ...;
  1, 5, 18, (37), 63, 97, 140, 193, 257, 333, 422, 525, 643, 777, ...;
  1, 6, 24, 87, (184), 324, 517, 774, 1107, 1529, 2054, 2697, 3474, ...;
  1, 7, 31, 118, 442, (959), 1733, 2840, 4369, 6423, 9120, 12594, ...;
  1, 8, 39, 157, 599, 2332, (5172), 9541, 15964, 25084, 37678, ...;
  1, 9, 48, 205, 804, 3136, 12677, (28641), 53725, 91403, 146077, ...;
  1, 10, 58, 263, 1067, 4203, 16880, 70605, (162008), 308085, ...;
  1, 11, 69, 332, 1399, 5602, 22482, 93087, 401172, (932503), ...;
  ...
For each row, remove the term along the diagonal (in parenthesis here),
and then take partial sums to obtain the next row.

Crossrefs

Cf. Diagonals: A007857, A130524, A130525; related: A000108, A001764.

Programs

  • PARI
    {T(n,k) = if(n<0||k<0,0,if(n==0,1,if(n>k+1, T(n,k-1) + T(n-1,k), T(n,k-1) + T(n-1,k+1))))}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
  • PARI
    /* Using Formula for G.F.: */
{T(n,k) = local(m=max(n,k)+1,C,F,D); C=subst(Ser(vector(m,r,binomial(2*r-2,r-1)/r)),x,x*y); F=subst(Ser(vector(m,r,binomial(3*r-3,r-1)/(2*r-1))),x,x*y); D=1/(1-x*y*C*F-x*y*F^2);A=D*(1/(1-y*F) + x*C*F/(1-x*C)); polcoeff(polcoeff(A+O(x^m),n,x)+O(y^m),k,y)}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

Formula

G.f.: A(x,y) = D(x*y)*( 1/(1 - y*F(x*y)) + x*C(x*y)*F(x*y)/(1 - x*C(x*y)) ), where D(x) = 1/(1 - x*C(x)*F(x) - x*F(x)^2) is the g.f. of the main diagonal (A007857), C(x) = 1 + x*C(x)^2 is the g.f. of Catalan numbers (A000108) and F(x) = 1 + x*F(x)^3 is the g.f. of ternary numbers (A001764).

A130524 Diagonal immediately below the main diagonal of square array A130523.

Original entry on oeis.org

1, 4, 18, 87, 442, 2332, 12677, 70605, 401172, 2317683, 13578615, 80502845, 482140580, 2912954129, 17733375207, 108676158775, 669914021414, 4151053001800, 25841001981211, 161534820531068, 1013566626969398, 6381398103680604, 40301852983776764, 255249505209864803, 1620819715569894894
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2007

Keywords

Crossrefs

Cf. A130523; diagonals: A007857, A130525; related: A000108, A001764.

Programs

  • PARI
    {a(n) = my(C,F,D); C=Ser(vector(n+1,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(n+1,r,binomial(3*r-3,r-1)/(2*r-1))); D=1/(1-x*C*F-x*F^2); polcoef(C*D*F+x*O(x^n),n,x)}
for(n=0,25,print1(a(n),", "))

Formula

G.f.: A(x) = C(x)*D(x)*F(x), where D(x) = 1/(1 - x*C(x)*F(x) - x*F(x)^2) is the g.f. of the main diagonal (A007857), C(x) = 1 + x*C(x)^2 is the g.f. of Catalan numbers (A000108) and F(x) = 1 + x*F(x)^3 is the g.f. of ternary numbers (A001764).

Extensions

Edited and corrected by Paul D. Hanna, Jan 27 2025

A137571 Main diagonal of square array A137570.

Original entry on oeis.org

1, 2, 10, 60, 397, 2802, 20710, 158428, 1244413, 9980220, 81394123, 672998498, 5628741195, 47535483498, 404790717079, 3471892750622, 29966295451511, 260080708564964, 2268416956569463, 19872441881999354, 174783803353387498
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2008

Keywords

Comments

A variant is A007857, the number of independent sets in rooted plane trees on n nodes.

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 397*x^4 + 2802*x^5 +...;
A(x) = 1/(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, A137572, A137573; A007857 (variant); 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=1/(1-x*C*F^2-x*F^3);polcoeff(A+O(x^m),n,x)}

Formula

G.f. A(x) = 1/(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.

A130525 Diagonal immediately above the main diagonal of square array A130523.

Original entry on oeis.org

1, 3, 13, 63, 324, 1733, 9541, 53725, 308085, 1793528, 10574165, 63018105, 379061652, 2298508911, 14035748542, 86240951745, 532812883413, 3307967729867, 20627845299471, 129141164822496, 811394148828087, 5114638998643903, 32336393838083539, 205000199138736499, 1302892014385402691
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2007

Keywords

Crossrefs

Cf. A130523; diagonals: A007857, A130524; related: A000108, A001764.

Programs

  • PARI
    {a(n) = my(C,F,D); C=Ser(vector(n+1,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(n+1,r,binomial(3*r-3,r-1)/(2*r-1))); D=1/(1-x*C*F-x*F^2); polcoef(D*F+x*O(x^n),n,x)}
for(n=0,25,print1(a(n),", "))

Formula

G.f.: A(x) = D(x)*F(x), where D(x) = 1/(1 - x*C(x)*F(x) - x*F(x)^2) is the g.f. of the main diagonal (A007857), C(x) = 1 + x*C(x)^2 is the g.f. of Catalan numbers (A000108) and F(x) = 1 + x*F(x)^3 is the g.f. of ternary numbers (A001764).

Extensions

Edited and corrected by Paul D. Hanna, Jan 27 2025
Showing 1-4 of 4 results.

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