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A359720 T(n,k) = coefficient of x^n*y^k in A(x,y) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (y + x^n)^n * A(x,y)^n.

%I A359720 #9 Jan 14 2023 10:53:33
%S A359720 1,1,1,2,4,5,1,7,21,9,20,51,49,7,43,170,179,66,2,110,454,711,381,54,
%T A359720 262,1367,2390,1894,523,25,674,3776,8361,8070,3496,469,5,1684,11062,
%U A359720 27082,33093,19129,4602,269,4397,31054,89389,125983,93908,33211,4325,91,11320,89935,283170,470439,421762,200449,43062,2846,14
%N A359720 T(n,k) = coefficient of x^n*y^k in A(x,y) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (y + x^n)^n * A(x,y)^n.
%C A359720 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
%C A359720 The terms in row n start with index k = 0 to k = floor(2*n/3), for n >= 0.
%C A359720 A359721(n) = Sum_{k=0..floor(2*n/3)} T(n,k), for n >= 0 (row sums).
%C A359720 A357797(n) = Sum_{k=0..floor(2*n/3)} T(n,k)*2^k, for n >= 0.
%C A359720 A359723(n) = Sum_{k=0..floor(2*n/3)} T(n,k)*3^k, for n >= 0.
%C A359720 A359724(n) = Sum_{k=0..floor(2*n/3)} T(n,k)*4^k, for n >= 0.
%C A359720 A355357(n) = T(n,0), for n >= 0.
%C A359720 A359725(n) = T(n+2,1), for n >= 0.
%C A359720 A359726(n) = T(n+3,2), for n >= 0.
%C A359720 A000108(n) = T(3*n,2*n), for n >= 0.
%C A359720 A359722(n) = T(3*n+1,2*n), for n >= 0.
%C A359720 A097613(n+2) = T(3*n+2,2*n+1), for n >= 0.
%H A359720 Paul D. Hanna, <a href="/A359720/b359720.txt">Table of n, a(n) for n = 0..4920</a>
%F A359720 G.f.: A(x,y) = Sum_{n>=0} Sum_{k=0..floor(2*n/3)} T(n,k)*x^n*y^k may be described by the following.
%F A359720 (1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (y + x^n)^n * A(x,y)^n.
%F A359720 (2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 + y*x^n)^n * A(x,y)^n ).
%F A359720 T(3*n,2*n) = binomial(2*n+1,n)/(2*n+1) = A000108(n), n >= 0 (Catalan numbers).
%F A359720 T(3*n+2,2*n+1) = binomial(2*n+1,n+1) + binomial(2*n+2,n) = A097613(n+2), for n >= 0.
%e A359720 G.f.: A(x,y) = 1 + x*(1) + x^2*(1 + 2*y) + x^3*(4 + 5*y + y^2) + x^4*(7 + 21*y + 9*y^2) + x^5*(20 + 51*y + 49*y^2 + 7*y^3) + x^6*(43 + 170*y + 179*y^2 + 66*y^3 + 2*y^4) + x^7*(110 + 454*y + 711*y^2 + 381*y^3 + 54*y^4) + x^8*(262 + 1367*y + 2390*y^2 + 1894*y^3 + 523*y^4 + 25*y^5) + x^9*(674 + 3776*y + 8361*y^2 + 8070*y^3 + 3496*y^4 + 469*y^5 + 5*y^6) + x^10*(1684 + 11062*y + 27082*y^2 + 33093*y^3 + 19129*y^4 + 4602*y^5 + 269*y^6) + x^11*(4397 + 31054*y + 89389*y^2 + 125983*y^3 + 93908*y^4 + 33211*y^5 + 4325*y^6 + 91*y^7) + x^12*(11320 + 89935*y + 283170*y^2 + 470439*y^3 + 421762*y^4 + 200449*y^5 + 43062*y^6 + 2846*y^7 + 14*y^8) + x^13*(29938 + 254654*y + 905307*y^2 + 1683683*y^3 + 1798279*y^4 + 1072012*y^5 + 329533*y^6 + 41858*y^7 + 1254*y^8) + x^14*(78641 + 733725*y + 2825245*y^2 + 5954300*y^3 + 7287245*y^4 + 5277807*y^5 + 2131517*y^6 + 421554*y^7 + 30194*y^8 + 336*y^9 ) + x^15*(210044 + 2088612*y + 8854116*y^2 + 20499318*y^3 + 28639206*y^4 + 24326336*y^5 + 12274991*y^6 + 3370105*y^7 + 420102*y^8 + 15745*y^9 + 42*y^10) + ...
%e A359720 This irregular triangle of coefficients T(n,k) of x^n*y^k, for n >= 0, k = 0..[2*n/3], in g.f. A(x,y) begin:
%e A359720 n = 0: [1],
%e A359720 n = 1: [1],
%e A359720 n = 2: [1, 2],
%e A359720 n = 3: [4, 5, 1],
%e A359720 n = 4: [7, 21, 9],
%e A359720 n = 5: [20, 51, 49, 7],
%e A359720 n = 6: [43, 170, 179, 66, 2],
%e A359720 n = 7: [110, 454, 711, 381, 54],
%e A359720 n = 8: [262, 1367, 2390, 1894, 523, 25],
%e A359720 n = 9: [674, 3776, 8361, 8070, 3496, 469, 5],
%e A359720 n = 10: [1684, 11062, 27082, 33093, 19129, 4602, 269],
%e A359720 n = 11: [4397, 31054, 89389, 125983, 93908, 33211, 4325, 91],
%e A359720 n = 12: [11320, 89935, 283170, 470439, 421762, 200449, 43062, 2846, 14],
%e A359720 n = 13: [29938, 254654, 905307, 1683683, 1798279, 1072012, 329533, 41858, 1254],
%e A359720 n = 14: [78641, 733725, 2825245, 5954300, 7287245, 5277807, 2131517, 421554, 30194, 336],
%e A359720 n = 15: [210044, 2088612, 8854116, 20499318, 28639206, 24326336, 12274991, 3370105, 420102, 15745, 42], ...
%e A359720 ...
%e A359720 in which various sequences are found along columns and diagonals:
%e A359720 T(n,0) = A355357(n) = [1, 1, 1, 4, 7, 20, 43, 110, 262, 674, 1684, ...],
%e A359720 T(n+2,1) = A359725(n) = [2, 5, 21, 51, 170, 454, 1367, 3776, 11062, ...],
%e A359720 T(n+3,2) = A359726(n) = [1, 9, 49, 179, 711, 2390, 8361, 27082, 89389, ...],
%e A359720 T(3*n,2*n) = A000108(n) = [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...],
%e A359720 T(3*n+1,2*n) = A359722(n) = [1, 9, 54, 269, 1254, 5642, 24828, 107613, ...],
%e A359720 T(3*n+2,2*n+1) = A097613(n+2) = [2, 7, 25, 91, 336, 1254, 4719, 17875, ...].
%o A359720 (PARI) /* Print this irregular triangle */
%o A359720 {T(n,k) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A359720 A[#A] = polcoeff(x - sum(n=-#A-1, #A+1, (-1)^n * x^n * (y + x^n +x*O(x^#A) )^n * Ser(A)^n ), #A-1) );
%o A359720 polcoeff(A[n+1],k,y)}
%o A359720 for(n=0, 15, for(k=0, (2*n)\3, print1(T(n,k), ", "));print(""))
%Y A359720 Cf. A359721 (row sums), A357797 (y=2), A359723 (y=3), A359724 (y=4).
%Y A359720 Cf. A355357 (T(n,0)), A359725 (T(n+2,1)), A359726 (T(n+3,2)).
%Y A359720 Cf. A000108 (T(3*n,2*n)), A097613 (T(3*n+2,2*n+1)), A359722 (T(3*n+1,2*n)).
%K A359720 nonn,tabf
%O A359720 0,4
%A A359720 _Paul D. Hanna_, Jan 13 2023