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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.
3
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
1,5
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).
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)).
EXAMPLE
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).
PROG
(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), ", ")); )}
CROSSREFS
Cf. A192405 (antidiagonal sums), A192406 (main diagonal), A192407.
Sequence in context: A305882 A339285 A363043 * A373746 A291261 A294206
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 30 2011
STATUS
approved