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.

A192404 G.f. satisfies: A(x,y) = 1 + Sum_{n>=1} x^n*y*A(x,y)^n/(1 - y*A(x,y)^(2*n)), where A(x,y) = 1 + Sum_{n>=1,k>=1} T(n,k)*x^n*y^k; here the coefficients T(n,k) form a square array and are read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 7, 14, 10, 1, 1, 11, 31, 38, 17, 1, 1, 16, 61, 114, 91, 26, 1, 1, 22, 111, 291, 357, 196, 37, 1, 1, 29, 190, 656, 1131, 971, 384, 50, 1, 1, 37, 309, 1345, 3092, 3771, 2367, 694, 65, 1, 1, 46, 481, 2563, 7575, 12393, 11150, 5286, 1173, 82, 1
Offset: 1

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Author

Paul D. Hanna, Jun 30 2011

Keywords

Comments

Related q-series identity:
Sum_{n>=1} x^n*y*q^n/(1-y*q^(2*n)) = Sum_{n>=1} y^n*x*q^(2*n-1)/(1-x*q^(2*n-1)); here q=A(x,y).

Examples

			Let A = g.f. A(x,y), then A satisfies the following relations:
A = 1 + x*y*A/(1-y*A^2) + x^2*y*A^2/(1-y*A^4) + x^3*y*A^3/(1-y*A^6) +...
A = 1 + y*x*A/(1-x*A) + y^2*x*A^3/(1-x*A^3) + y^3*x*A^5/(1-x*A^5) +...
The square array of coefficients in A(x,y) begins:
[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,...];
[0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...];
[0,1,2,5,10,17,26,37,50,65,82,101,122,145,170,197,226,257,...];
[0,1,4,14,38,91,196,384,694,1173,1876,2866,4214,5999,8308,...];
[0,1,7,31,114,357,971,2367,5286,10969,21367,39391,69202,...];
[0,1,11,61,291,1131,3771,11150,29828,73329,167767,360791,...];
[0,1,16,111,656,3092,12393,43464,136434,390259,1031194,...];
[0,1,22,190,1345,7575,35839,146712,533050,1751371,5278494,...];
[0,1,29,309,2563,17011,93599,441925,1838094,6865479,...];
[0,1,37,481,4609,35563,224947,1212807,5721008,24088842,...];
[0,1,46,721,7906,70021,504448,3078603,16340045,77009425,...]; ...
in which the zeroth row and column are omitted from this sequence.
EXPLICIT EXPANSIONS of the generating function are as follows.
SUMMATION ALONG ROWS gives:
A(x,y) = 1 + x*y/(1-y) + x^2*(y - y^2 + 2*y^3)/(1-y)^3 + x^3*(y - y^2 + 4*y^3 - 2*y^4 + 6*y^5)/(1-y)^5 + x^4*(y + 3*y^3 + 9*y^4 + 5*y^6 + 22*y^7)/(1-y)^7 + x^5*(y + 2*y^2 - 2*y^3 + 54*y^4 - 90*y^5 + 204*y^6 - 133*y^7 + 98*y^8 + 90*y^9)/(1-y)^9 +...
in which the denominator polynomials are the odd powers of (1-y).
The coefficients of y in the above numerator polynomials begin:
[1];
[1,-1,2];
[1,-1,4,-2,6];
[1,0,3,9,0,5,22];
[1,2,-2,54,-90,204,-133,98,90];
[1,5,-10,150,-329,964,-1339,2025,-1385,868,394];
[1,9,-18,305,-667,2377,-3763,7967,-10012,14378,-10323,6388,1806];
[1,14,-21,518,-819,3536,-2367,6387,-1660,20583,-35708,77417,-64888,43361,8558];...
in which the row sums form A052701 (related to Catalan numbers), and the rightmost border forms the large Schroeder numbers (A006318).
SUMMATION ALONG COLUMNS gives:
A(x,y) = 1 + y*x/(1-x) + y^2*(x - x^2 + x^3)/(1-x)^3 + y^3*(x - x^3 + x^4 + x^5)/(1-x)^5 + y^4*(x + 3*x^2 - 11*x^3 + 23*x^4 - 24*x^5 + 12*x^6 + x^7)/(1-x)^7 + y^5*(x + 8*x^2 - 26*x^3 + 66*x^4 - 108*x^5 + 137*x^6 - 117*x^7 + 52*x^8 + x^9)/(1-x)^9 ...
in which the denominator polynomials are the odd powers of (1-x).
The coefficients of x in the above numerator polynomials begin:
[1];
[1,-1,1];
[1,0,-1,1,1];
[1,3,-11,23,-24,12,1];
[1,8,-26,66,-108,137,-117,52,1];
[1,15,-35,80,-90,95,-164,330,-377,186,1];
[1,24,-19,-25,464,-1516,3075,-4066,3274,-928,-778,625,1];
[1,35,49,-329,2023,-6479,15515,-28703,41895,-46744,36552,-15870,429,2054,1];...
in which the row sums form the Catalan numbers (A000108).

Crossrefs

Cf. A192405 (antidiagonal sums), A192406 (main diagonal), A192407.

Programs

  • PARI
    {T(n,k)=local(A=1+x*y);for(i=1,n+k,A=1+sum(m=1,n+k,x^m*y*A^m/(1-y*A^(2*m)+x*O(x^n)+y*O(y^k))));polcoeff(polcoeff(A,n,x),k,y)}
  • PARI
    {T(n,k)=local(A=1+x*y);for(i=1,n+k,A=1+sum(m=1,n+k,y^m*x*A^(2*m-1)/(1-x*A^(2*m-1)+x*O(x^n)+y*O(y^k))));polcoeff(polcoeff(A,k,y),n,x)}
/* Print the coefficients as a square array: */
{for(n=1,12,for(k=1,18-n,print1(T(n,k),","));print(""))}
/* Print the array in flattened format: */
{for(n=1,12,for(k=1,n,print1(T(n-k+1,k),","));)}

Formula

G.f. satisfies: A(x,y) = 1 + Sum_{n>=1} y^n*x*A(x,y)^(2*n-1)/(1 - x*A(x,y)^(2*n-1)).

A192406 Main diagonal of square array A192404, with a(0)=1.

Original entry on oeis.org

1, 1, 2, 14, 114, 1131, 12393, 146712, 1838094, 24088842, 327526513, 4593918125, 66198455671, 977113573208, 14741071612583, 226941948201964, 3561383719180100, 56926946565867437, 926444637518092848, 15347533201937448776, 258809102457332568964
Offset: 0

Views

Author

Paul D. Hanna, Jun 30 2011

Keywords

Comments

The g.f. G(x,y) of square array A192404 satisfies the relations:
G(x,y) = 1 + Sum{n>=1} x^n*y*G(x,y)^n/(1 - y*G(x,y)^(2*n)),
G(x,y) = 1 + Sum{n>=1} y^n*x*G(x,y)^(2*n-1)/(1 - x*G(x,y)^(2*n-1)),
where G(x,y) = 1 + Sum_{n>=1,k>=1} A192404(n,k)*x^n*y^k, and this sequence consists of the diagonal terms a(n) = A192404(n,n); what then is the g.f. of this sequence?

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 14*x^3 + 114*x^4 + 1131*x^5 + 12393*x^6 +...

Crossrefs

Cf. A192404, A192405, A192407.

Programs

  • PARI
    {a(n)=local(A=1+x*y);for(i=1,n,A=1+sum(m=1,n,x^m*y*A^m/(1-y*A^(2*m)+x*O(x^n)+y*O(y^n))));polcoeff(polcoeff(A,n,x),n,y)}
  • PARI
    {a(n)=local(A=1+x*y);for(i=1,n,A=1+sum(m=1,n,y^m*x*A^(2*m-1)/(1-x*A^(2*m-1)+x*O(x^n)+y*O(y^n))));polcoeff(polcoeff(A,n,y),n,x)}

A192407 A diagonal of square array A192404.

Original entry on oeis.org

1, 4, 31, 291, 3092, 35839, 441925, 5721008, 77009425, 1071034612, 15319883964, 224628789200, 3368096726910, 51552652046550, 804490751228163, 12788591015038781, 206977224029107906, 3409582505289727239, 57165456138722305360
Offset: 1

Views

Author

Paul D. Hanna, Jun 30 2011

Keywords

Comments

The g.f. G(x,y) of square array A192404 satisfies the relations:
G(x,y) = 1 + Sum{n>=1} x^n*y*G(x,y)^n/(1 - y*G(x,y)^(2*n)),
G(x,y) = 1 + Sum{n>=1} y^n*x*G(x,y)^(2*n-1)/(1 - x*G(x,y)^(2*n-1)),
where G(x,y) = 1 + Sum_{n>=1,k>=1} A192404(n,k)*x^n*y^k, and this sequence consists of the diagonal terms a(n) = A192404(n+1,n) for n>=1.

Examples

			G.f.: A(x) = x + 4*x^2 + 31*x^3 + 291*x^4 + 3092*x^5 + 35839*x^6 +...

Crossrefs

Cf. A192404, A192405, A192406.

Programs

  • PARI
    {a(n)=local(A=x*y);for(i=1,n+1,A=1+sum(m=1,n+1,x^m*y*A^m/(1-y*A^(2*m)+x*O(x^n)+y*O(y^n))));polcoeff(polcoeff(A,n+1,x),n,y)}
  • PARI
    {a(n)=local(A=x*y);for(i=1,n+1,A=1+sum(m=1,n+1,y^m*x*A^(2*m-1)/(1-x*A^(2*m-1)+x*O(x^n)+y*O(y^n))));polcoeff(polcoeff(A,n,y),n+1,x)}

A192399 G.f. A(x) satisfies: A(x) = 1 + Sum_{n>=1} x^n * A(x)^n/(1 - x*A(x)^(2*n)).

Original entry on oeis.org

1, 1, 3, 11, 48, 233, 1218, 6722, 38668, 229864, 1403618, 8766186, 55818141, 361499355, 2376956264, 15845876429, 106988044753, 731026642533, 5051920683481, 35296182297157, 249249589433312, 1778775804736254, 12828718640894604
Offset: 0

Views

Author

Paul D. Hanna, Jun 30 2011

Keywords

Comments

Related q-series identity:
Sum_{n>=1} z^n*y*q^n/(1-y*q^(2*n)) = Sum_{n>=1} y^n*z*q^(2*n-1)/(1-z*q^(2*n-1)); here q=A(x), y=x, z=x.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 48*x^4 + 233*x^5 + 1218*x^6 +...
which satisfies the following relations:
A(x) = 1 + x*A(x)/(1-x*A(x)^2) + x^2*A(x)^2/(1-x*A(x)^4) + x^3*A(x)^3/(1-x*A(x)^6) +...
A(x) = 1 + x*A(x)/(1-x*A(x)) + x^2*A(x)^3/(1-x*A(x)^3) + x^3*A(x)^5/(1-x*A(x)^5) +...
A(x) = 1 + x*A(x) + x^2*A(x)^3*(1 + 1/A(x)) + x^3*A(x)^6*(1 + 1/A(x) + 1/A(x)^3) + x^4*A(x)^10*(1 + 1/A(x) + 1/A(x)^3 + 1/A(x)^6) +...

Crossrefs

Cf. A192400, A192401, A192405.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*A^m/(1-x*A^(2*m)+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*A^(2*m-1)/(1-x*A^(2*m-1)+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*A^(m*(m+1)/2)*sum(k=0,m-1,(A+x*O(x^n))^(-k*(k+1)/2) ) ) );polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^(2*n-1)/(1 - x*A(x)^(2*n-1)).
G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^(n*(n+1)/2) * Sum_{k=0..n-1} A(x)^(-k*(k+1)/2). - Paul D. Hanna, Jul 01 2011
Showing 1-4 of 4 results.

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