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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

%I

%S 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,

%T 21,35,35,21,7,8,15,8,36,56,90,56,36,8,15,13,9,36,93,126,126,93,36,9,

%U 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

%N 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.

%H Paul D. Hanna, <a href="/A322200/b322200.txt">Table of n, a(n) for n = 0..1890</a>

%F Sum_{k=0..n} T(n-k,k) = A054598(n) = Sum_{d|n} d*2^(n/d).

%F Sum_{k=0..n} T(n-k,k) * k/n = A054599(n) = Sum_{d|n} d*2^(n/d - 1).

%F 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.

%F FORMULAS FOR TERMS.

%F T(n,k) = T(k,n) for n >= 0, k >= 0.

%F T(0,0) = 0.

%F T(n,0) = sigma(n) for n > 0.

%F T(0,k) = sigma(k) for n > 0.

%F T(n,1) = n+1, for n >= 0.

%F T(1,k) = k+1, for k >= 0.

%F T(2*n,2) = T(2*n+1,2) = (n+1)*(2*n+3).

%F T(2,2*k) = T(2,2*k+1) = (k+1)*(2*k+3).

%F COLUMN GENERATING FUNCTIONS.

%F Row 0: log(P(x)), where P(x) = Product_{n>=1} 1/(1 - x^n).

%F Row 1: 1/(1-x)^2.

%F Row 2: (3 + x^2)/((1-x)*(1-x^2)^2).

%F Row 3: (4 - 4*x + 6*x^2 + 2*x^3 + x^4)/((1-x)^2*(1-x^3)^2).

%F 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).

%F 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).

%F 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).

%e 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 + ...

%e such that

%e exp( L(x,y) ) = Product_{n>=1} 1/(1 - (x^n + y^n)), or

%e L(x,y) = Sum_{n>=1} -log(1 - (x^n + y^n)),

%e where

%e L(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n+k),

%e in which the constant term is taken to be zero: L(0,0) = 0.

%e SQUARE TABLE.

%e The square table of coefficients T(n,k) of x^n*y^k/(n+k) in L(x,y) begins

%e 0, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, ...;

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...;

%e 3, 3, 10, 10, 21, 21, 36, 36, 55, 55, 78, 78, 105, ...;

%e 4, 4, 10, 26, 35, 56, 93, 120, 165, 232, 286, 364, ...;

%e 7, 5, 21, 35, 90, 126, 230, 330, 537, 715, 1043, 1365, ...;

%e 6, 6, 21, 56, 126, 262, 462, 792, 1287, 2002, 3018, ...;

%e 12, 7, 36, 93, 230, 462, 994, 1716, 3073, 5035, 8120, ...;

%e 8, 8, 36, 120, 330, 792, 1716, 3446, 6435, 11440, 19448, ...;

%e 15, 9, 55, 165, 537, 1287, 3073, 6435, 13050, 24310, 44010, ...;

%e 13, 10, 55, 232, 715, 2002, 5035, 11440, 24310, 48698, 92378, ...;

%e 18, 11, 78, 286, 1043, 3018, 8120, 19448, 44010, 92378, 185310, ...;

%e 12, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, ...; ...

%e TRIANGLE.

%e Alternatively, this sequence may be written as a triangle, starting as

%e 0;

%e 1, 1;

%e 3, 2, 3;

%e 4, 3, 3, 4;

%e 7, 4, 10, 4, 7;

%e 6, 5, 10, 10, 5, 6;

%e 12, 6, 21, 26, 21, 6, 12;

%e 8, 7, 21, 35, 35, 21, 7, 8;

%e 15, 8, 36, 56, 90, 56, 36, 8, 15;

%e 13, 9, 36, 93, 126, 126, 93, 36, 9, 13;

%e 18, 10, 55, 120, 230, 262, 230, 120, 55, 10, 18;

%e 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12;

%e 28, 12, 78, 232, 537, 792, 994, 792, 537, 232, 78, 12, 28;

%e 14, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 14;

%e 24, 14, 105, 364, 1043, 2002, 3073, 3446, 3073, 2002, 1043, 364, 105, 14, 24;

%e 24, 15, 105, 470, 1365, 3018, 5035, 6435, 6435, 5035, 3018, 1365, 470, 105, 15, 24;

%e 31, 16, 136, 560, 1892, 4368, 8120, 11440, 13050, 11440, 8120, 4368, 1892, 560, 136, 16, 31; ...

%e where L(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n-k,k)*x^(n-k)*y^k / n.

%o (PARI)

%o {L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}

%o {T(n,k) = polcoeff( (n+k)*polcoeff( L,n,x),k,y)}

%o for(n=0,16, for(k=0,16, print1( T(n,k),", ") );print(""))

%Y Cf. A322210 (exp), A322201 (main diagonal), A322203, A322205, A322207, A322209.

%Y Cf. A054598 (antidiagonal sums), A054599.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Nov 30 2018

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Last modified April 23 01:31 EDT 2021. Contains 343198 sequences. (Running on oeis4.)