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.
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
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).
Programs
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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)).
Comments