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 A355360 G.f. A(x,y) satisfies: x*y*A(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. 8
 1, 0, 1, 0, 3, 2, 0, 9, 12, 5, 0, 22, 54, 46, 14, 0, 51, 196, 282, 175, 42, 0, 108, 630, 1360, 1365, 666, 132, 0, 221, 1836, 5635, 8190, 6321, 2541, 429, 0, 429, 4984, 20850, 41405, 45326, 28448, 9724, 1430, 0, 810, 12744, 70737, 184527, 270060, 237209, 125532, 37323, 4862, 0, 1479, 31050, 223652, 745745, 1404102, 1625932, 1193116, 546039, 143650, 16796 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The main diagonal equals A000108, the Catalan numbers. Conjectures. (C.1) Column 1 equals A000716, the number of partitions into parts of 3 kinds. (C.2) Column 2 equals twice A023005, the number of partitions into parts of 6 kinds. 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. A355361(n) = Sum_{k=0..n} T(n,k) for n >= 0 (row sums). A355362(n) = Sum_{k=0..n} T(n,k) * 2^k for n >= 0. A355363(n) = Sum_{k=0..n} T(n,k) * 3^k for n >= 0. A355364(n) = Sum_{k=0..floor(n/2)} T(n-k,k) for n >= 0 (antidiagonal sums). A355365(n) = T(2*n,n) for n >= 0 (central terms of this triangle). LINKS Paul D. Hanna, Table of n, a(n) for n = 0..5150 FORMULA G.f. A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*y^k satisfies: (1) x*y*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n. (2) -x*y*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x,y)^n. (3) x*y*A(x)*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. (4) A(x,y) = B(x, y*A(x,y)) and A(x, y/B(x,y)) = B(x,y) where x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * B(x,y)^n, and B(x,y) is the g.f. of table A355350. EXAMPLE G.f.: A(x,y) = 1 + x*y + x^2*(3*y + 2*y^2) + x^3*(9*y + 12*y^2 + 5*y^3) + x^4*(22*y + 54*y^2 + 46*y^3 + 14*y^4) + x^5*(51*y + 196*y^2 + 282*y^3 + 175*y^4 + 42*y^5) + x^6*(108*y + 630*y^2 + 1360*y^3 + 1365*y^4 + 666*y^5 + 132*y^6) + x^7*(221*y + 1836*y^2 + 5635*y^3 + 8190*y^4 + 6321*y^5 + 2541*y^6 + 429*y^7) + x^8*(429*y + 4984*y^2 + 20850*y^3 + 41405*y^4 + 45326*y^5 + 28448*y^6 + 9724*y^7 + 1430*y^8) + x^9*(810*y + 12744*y^2 + 70737*y^3 + 184527*y^4 + 270060*y^5 + 237209*y^6 + 125532*y^7 + 37323*y^8 + 4862*y^9) + x^10*(1479*y + 31050*y^2 + 223652*y^3 + 745745*y^4 + 1404102*y^5 + 1625932*y^6 + 1193116*y^7 + 546039*y^8 + 143650*y^9 + 16796*y^10) + ... where x*y*A(x) = ... - 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 + ... also, x*y*A(x)*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)) * ... TRIANGLE. The triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for k = 0..n in row n, begins: n=0: [1]; n=1: [0, 1]; n=2: [0, 3, 2]; n=3: [0, 9, 12, 5]; n=4: [0, 22, 54, 46, 14]; n=5: [0, 51, 196, 282, 175, 42]; n=6: [0, 108, 630, 1360, 1365, 666, 132]; n=7: [0, 221, 1836, 5635, 8190, 6321, 2541, 429]; n=8: [0, 429, 4984, 20850, 41405, 45326, 28448, 9724, 1430]; n=9: [0, 810, 12744, 70737, 184527, 270060, 237209, 125532, 37323, 4862]; n=10: [0, 1479, 31050, 223652, 745745, 1404102, 1625932, 1193116, 546039, 143650, 16796]; n=11: [0, 2640, 72560, 667005, 2784110, 6565030, 9646462, 9242178, 5826171, 2349490, 554268, 58786]; n=12: [0, 4599, 163632, 1892670, 9729720, 28161819, 51126740, 61555824, 50308245, 27806065, 10023948, 2143428, 208012]; ... in which column 1 appears to equal A000716, the coefficients in P(x)^3, and column 2 appears to equal twice A023005, the coefficients in P(x)^6, where P(x) is the partition function and begins 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 + ... Also, the power series expansions of P(x)^3 and P(x)^6 begin 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 + ... 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 + ... The main diagonal equals the Catalan numbers (A000108), where g.f. C(x) = 1 + x*C(x)^2 begins C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + ... + A000108(n)*x^n + ... PROG (PARI) {T(n, k) = my(A=[1, y], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*(#A)+9)); A[#A] = polcoeff( x*y*Ser(A) - sum(m=-t, t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); polcoeff(A[n+1], k, y)} for(n=0, 12, for(k=0, n, print1( T(n, k), ", ")); print("")) CROSSREFS Cf. A000108 (main diagonal), A000041, A000716, A023005. Cf. A355361 (y=1), A355362 (y=2), A355363 (y=3), A355364, A355365. Cf. A355350 (related table). Sequence in context: A248181 A229728 A077907 * A067346 A360282 A356654 Adjacent sequences: A355357 A355358 A355359 * A355361 A355362 A355363 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Jul 19 2022 STATUS approved

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Last modified September 8 22:43 EDT 2024. Contains 375759 sequences. (Running on oeis4.)