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 A322200 L.g.f.: L(x,y) = log( Product_{n>=1} 1/(1 - (x^n + y^n)) ), where L(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k / (n+k) such that L(0,0) = 0, as a symmetric square table of coefficients T(n,k) read by antidiagonals starting with T(0,0) = 0. 13
 0, 1, 1, 3, 2, 3, 4, 3, 3, 4, 7, 4, 10, 4, 7, 6, 5, 10, 10, 5, 6, 12, 6, 21, 26, 21, 6, 12, 8, 7, 21, 35, 35, 21, 7, 8, 15, 8, 36, 56, 90, 56, 36, 8, 15, 13, 9, 36, 93, 126, 126, 93, 36, 9, 13, 18, 10, 55, 120, 230, 262, 230, 120, 55, 10, 18, 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12, 28, 12, 78, 232, 537, 792, 994, 792, 537, 232, 78, 12, 28, 14, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 14, 24, 14, 105, 364, 1043, 2002, 3073, 3446, 3073, 2002, 1043, 364, 105, 14, 24, 24, 15, 105, 470, 1365, 3018, 5035, 6435, 6435, 5035, 3018, 1365, 470, 105, 15, 24, 31, 16, 136, 560, 1892, 4368, 8120, 11440, 13050, 11440, 8120, 4368, 1892, 560, 136, 16, 31 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..1890 FORMULA Sum_{k=0..n} T(n-k,k) = A054598(n) = Sum_{d|n} d*2^(n/d). Sum_{k=0..n} T(n-k,k) * k/n = A054599(n) = Sum_{d|n} d*2^(n/d - 1). Sum_{k=0..n} T(n-k,k) * 2^k = A322209(n) = [x^n] log( Product_{k>=1} 1/(1 - (2^k+1)*x^k) ) for n >= 0. FORMULAS FOR TERMS. T(n,k) = T(k,n) for n >= 0, k >= 0. T(0,0) = 0. T(n,0) = sigma(n) for n > 0. T(0,k) = sigma(k) for n > 0. T(n,1) = n+1, for n >= 0. T(1,k) = k+1, for k >= 0. T(2*n,2) = T(2*n+1,2) = (n+1)*(2*n+3). T(2,2*k) = T(2,2*k+1) = (k+1)*(2*k+3). COLUMN GENERATING FUNCTIONS. Row 0: log(P(x)), where P(x) = Product_{n>=1} 1/(1 - x^n). Row 1: 1/(1-x)^2. Row 2: (3 + x^2)/((1-x)*(1-x^2)^2). Row 3: (4 - 4*x + 6*x^2 + 2*x^3 + x^4)/((1-x)^2*(1-x^3)^2). Row 4: (7 - 9*x + 11*x^2 + 7*x^3 + 9*x^4 + x^5 + 5*x^6 + x^7)/((1-x)^2*(1-x^2)*(1-x^4)^2). Row 5: (6 - 18*x + 33*x^2 - 16*x^3 + 10*x^4 + 4*x^5 + 3*x^6 + 2*x^7 + x^8)/((1-x)^3*(1-x^5)^2). Row 6: (12 - 41*x + 56*x^2 + 13*x^3 - 49*x^4 - 20*x^5 + 105*x^6 - 126*x^7 + 85*x^8 - 62*x^9 + 24*x^10 - 28*x^11 + 39*x^12 - 25*x^13 + 15*x^14 + x^15 + x^16) / ((1-x)^4*(1-x^2)^2*(1-x^3)*(1-x^6)^2). EXAMPLE L.g.f.: L(x,y) = (x + y)/1 + (3*x^2 + 2*x*y + 3*y^2)/2 + (4*x^3 + 3*x^2*y + 3*x*y^2 + 4*y^3)/3 + (7*x^4 + 4*x^3*y + 10*x^2*y^2 + 4*x*y^3 + 7*y^4)/4 + (6*x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + 6*y^5)/5 + (12*x^6 + 6*x^5*y + 21*x^4*y^2 + 26*x^3*y^3 + 21*x^2*y^4 + 6*x*y^5 + 12*y^6)/6 + (8*x^7 + 7*x^6*y + 21*x^5*y^2 + 35*x^4*y^3 + 35*x^3*y^4 + 21*x^2*y^5 + 7*x*y^6 + 8*y^7)/7 + (15*x^8 + 8*x^7*y + 36*x^6*y^2 + 56*x^5*y^3 + 90*x^4*y^4 + 56*x^3*y^5 + 36*x^2*y^6 + 8*x*y^7 + 15*y^8)/8 + ... such that exp( L(x,y) ) = Product_{n>=1} 1/(1 - (x^n + y^n)), or L(x,y) = Sum_{n>=1} -log(1 - (x^n + y^n)), where L(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n+k), in which the constant term is taken to be zero: L(0,0) = 0. SQUARE TABLE. The square table of coefficients T(n,k) of x^n*y^k/(n+k) in L(x,y) begins 0, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, ...; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...; 3, 3, 10, 10, 21, 21, 36, 36, 55, 55, 78, 78, 105, ...; 4, 4, 10, 26, 35, 56, 93, 120, 165, 232, 286, 364, ...; 7, 5, 21, 35, 90, 126, 230, 330, 537, 715, 1043, 1365, ...; 6, 6, 21, 56, 126, 262, 462, 792, 1287, 2002, 3018, ...; 12, 7, 36, 93, 230, 462, 994, 1716, 3073, 5035, 8120, ...; 8, 8, 36, 120, 330, 792, 1716, 3446, 6435, 11440, 19448, ...; 15, 9, 55, 165, 537, 1287, 3073, 6435, 13050, 24310, 44010, ...; 13, 10, 55, 232, 715, 2002, 5035, 11440, 24310, 48698, 92378, ...; 18, 11, 78, 286, 1043, 3018, 8120, 19448, 44010, 92378, 185310, ...; 12, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, ...; ... TRIANGLE. Alternatively, this sequence may be written as a triangle, starting as 0; 1, 1; 3, 2, 3; 4, 3, 3, 4; 7, 4, 10, 4, 7; 6, 5, 10, 10, 5, 6; 12, 6, 21, 26, 21, 6, 12; 8, 7, 21, 35, 35, 21, 7, 8; 15, 8, 36, 56, 90, 56, 36, 8, 15; 13, 9, 36, 93, 126, 126, 93, 36, 9, 13; 18, 10, 55, 120, 230, 262, 230, 120, 55, 10, 18; 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12; 28, 12, 78, 232, 537, 792, 994, 792, 537, 232, 78, 12, 28; 14, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 14; 24, 14, 105, 364, 1043, 2002, 3073, 3446, 3073, 2002, 1043, 364, 105, 14, 24; 24, 15, 105, 470, 1365, 3018, 5035, 6435, 6435, 5035, 3018, 1365, 470, 105, 15, 24; 31, 16, 136, 560, 1892, 4368, 8120, 11440, 13050, 11440, 8120, 4368, 1892, 560, 136, 16, 31; ... where L(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n-k,k)*x^(n-k)*y^k / n. PROG (PARI) {L = sum(n=1, 61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) ); } {T(n, k) = polcoeff( (n+k)*polcoeff( L, n, x), k, y)} for(n=0, 16, for(k=0, 16, print1( T(n, k), ", ") ); print("")) CROSSREFS Cf. A322210 (exp), A322201 (main diagonal), A322203, A322205, A322207, A322209. Cf. A054598 (antidiagonal sums), A054599. Sequence in context: A267376 A267380 A217287 * A028292 A256244 A233386 Adjacent sequences: A322197 A322198 A322199 * A322201 A322202 A322203 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Nov 30 2018 STATUS approved

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Last modified September 18 15:02 EDT 2024. Contains 376000 sequences. (Running on oeis4.)