A355350 - OEIS

%I #7 Jun 30 2022 10:39:21

%S 1,0,1,0,3,1,0,9,6,1,0,22,27,10,1,0,51,98,66,15,1,0,108,315,340,135,

%T 21,1,0,221,918,1495,910,246,28,1,0,429,2492,5838,5070,2086,413,36,1,

%U 0,810,6372,20805,24543,14280,4284,652,45,1,0,1479,15525,68816,106535,83559,35168,8100,981,55,1,0,2640,36280,213945,423390,432930,243208,78282,14355,1420,66,1

%N G.f. A(x,y) satisfies: x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.

%C The term T(n,k) is found in row n and column k of this triangle, and can be used to derive the following sequences.

%C A355351(n) = Sum_{k=0..n} T(n,k) for n >= 0 (row sums).

%C A355352(n) = Sum_{k=0..n} T(n,k) * 2^k for n >= 0.

%C A355353(n) = Sum_{k=0..n} T(n,k) * 3^k for n >= 0.

%C A355354(n) = Sum_{k=0..n} T(n,k) * 4^k for n >= 0.

%C A355355(n) = Sum_{k=0..n} T(n,k) * 5^k for n >= 0.

%C A355356(n) = Sum_{k=0..floor(n/2)} T(n-k,k) for n >= 0 (antidiagonal sums).

%C A355357(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) for n >= 0.

%C A354658(n) = T(2*n,n) for n >= 0 (central terms of this triangle).

%C Conjectures:

%C (C.1) Column 1 equals A000716, the number of partitions into parts of 3 kinds;

%C (C.2) Column 2 equals A023005, the number of partitions into parts of 6 kinds.

%H Paul D. Hanna, <a href="/A355350/b355350.txt">Table of n, a(n) for n = 0..5150</a>

%F G.f. A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*y^k satisfies:

%F (1) x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.

%F (2) x*y*P(x) = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

%e G.f.: A(x,y) = 1 + x*y + x^2*(3*y + y^2) + x^3*(9*y + 6*y^2 + y^3) + x^4*(22*y + 27*y^2 + 10*y^3 + y^4) + x^5*(51*y + 98*y^2 + 66*y^3 + 15*y^4 + y^5) + x^6*(108*y + 315*y^2 + 340*y^3 + 135*y^4 + 21*y^5 + y^6) + x^7*(221*y + 918*y^2 + 1495*y^3 + 910*y^4 + 246*y^5 + 28*y^6 + y^7) + x^8*(429*y + 2492*y^2 + 5838*y^3 + 5070*y^4 + 2086*y^5 + 413*y^6 + 36*y^7 + y^8) + x^9*(810*y + 6372*y^2 + 20805*y^3 + 24543*y^4 + 14280*y^5 + 4284*y^6 + 652*y^7 + 45*y^8 + y^9) + x^10*(1479*y + 15525*y^2 + 68816*y^3 + 106535*y^4 + 83559*y^5 + 35168*y^6 + 8100*y^7 + 981*y^8 + 55*y^9 + y^10) + ...

%e where

%e x*y = ... - x^10/A(x,y)^5 + x^6/A(x,y)^4 - x^3/A(x,y)^3 + x/A(x,y)^2 - 1/A(x,y) + 1 - x*A(x,y) + x^3*A(x,y)^2 - x^6*A(x,y)^3 + x^10*A(x,y)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x,y)^n + ...

%e also, given P(x) is the partition function (A000041),

%e x*y*P(x) = (1 - x*A(x,y))*(1 - 1/A(x,y)) * (1 - x^2*A(x,y))*(1 - x/A(x,y)) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y)) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y)) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y)) * ...

%e TRIANGLE.

%e The triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for k = 0..n in row n, begins:

%e n=0: [1];

%e n=1: [0, 1];

%e n=2: [0, 3, 1];

%e n=3: [0, 9, 6, 1];

%e n=4: [0, 22, 27, 10, 1];

%e n=5: [0, 51, 98, 66, 15, 1];

%e n=6: [0, 108, 315, 340, 135, 21, 1];

%e n=7: [0, 221, 918, 1495, 910, 246, 28, 1];

%e n=8: [0, 429, 2492, 5838, 5070, 2086, 413, 36, 1];

%e n=9: [0, 810, 6372, 20805, 24543, 14280, 4284, 652, 45, 1];

%e n=10: [0, 1479, 15525, 68816, 106535, 83559, 35168, 8100, 981, 55, 1];

%e n=11: [0, 2640, 36280, 213945, 423390, 432930, 243208, 78282, 14355, 1420, 66, 1];

%e n=12: [0, 4599, 81816, 630890, 1563705, 2033244, 1472261, 629280, 160965, 24145, 1991, 78, 1];

%e ...

%e in which column 1 appears to equal A000716, the coefficients in P(x)^3,

%e and column 2 appears to equal A023005, the coefficients in P(x)^6,

%e where P(x) is the partition function and begins

%e P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + ... + A000041(n)*x^n + ...

%e Also, the power series expansions of P(x)^3 and P(x)^6 begin

%e P(x)^3 = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 108*x^5 + 221*x^6 + 429*x^7 + 810*x^8 + 1479*x^9 + 2640*x^10 + ... + A000716(n)*x^n + ...

%e P(x)^6 = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2492*x^6 + 6372*x^7 + 15525*x^8 + 36280*x^9 + 81816*x^10 + ... + A023005(n)*x^n + ...

%o (PARI) {T(n,k) = my(A=[1,y],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));

%o A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));polcoeff(A[n+1],k,y)}

%o for(n=0,12, for(k=0,n, print1( T(n,k),", "));print(""))

%Y Cf. A355351 (row sums), A355352, A355353, A355354, A355355.

%Y Cf. A355356, A355357, A354658 (central terms).

%Y Cf. A354645, A354650 (related table), A000041, A000716, A023005.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Jun 29 2022