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 A322210 G.f.: P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)), where P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k, as  a square table of coefficients T(n,k) read by antidiagonals. 7

%I

%S 1,1,1,2,2,2,3,4,4,3,5,7,10,7,5,7,12,18,18,12,7,11,19,34,38,34,19,11,

%T 15,30,56,74,74,56,30,15,22,45,94,133,158,133,94,45,22,30,67,146,233,

%U 297,297,233,146,67,30,42,97,228,385,550,602,550,385,228,97,42,56,139,340,623,951,1166,1166,951,623,340,139,56

%N G.f.: P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)), where P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k, as a square table of coefficients T(n,k) read by antidiagonals.

%C Conjecture: T(n,k) is the number of ways to form the subsum k from the partitions of n, where n and k are integers such that 0 <= k <= n. For example, t(4,2)=10; the five partitions of 4 are (4), (3,1), (2,2), (2,1,1), (1,1,1,1) with subsum 2 occurring {0,0,2,2,6) times for a total of 10. - _George Beck_, Jan 03 2020

%H Alois P. Heinz, <a href="/A322210/b322210.txt">Antidiagonals n = 0..200</a> (first 61 antidiagonals from Paul D. Hanna)

%F FORMULAS FOR TERMS.

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

%F T(n,0) = A000041(n) for n >= 0, where A000041 is the partition numbers.

%F T(n,1) = A000070(n) for n >= 0, where A000070 is the sum of partitions.

%F ROW GENERATING FUNCTIONS.

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

%F Row 1: 1/( (1-x) * Product_{n>=1} (1 - x^n) ).

%F Row 2: 2/( (1-x) * (1-x^2) * Product_{n>=1} (1 - x^n) ).

%e G.f.: P(x,y) = 1 + (x + y) + (2*x^2 + 2*x*y + 2*y^2) + (3*x^3 + 4*x^2*y + 4*x*y^2 + 3*y^3) + (5*x^4 + 7*x^3*y + 10*x^2*y^2 + 7*x*y^3 + 5*y^4) + (7*x^5 + 12*x^4*y + 18*x^3*y^2 + 18*x^2*y^3 + 12*x*y^4 + 7*y^5) + (11*x^6 +19*x^5*y + 34*x^4*y^2 + 38*x^3*y^3 + 34*x^2*y^4 + 19*x*y^5 + 11*y^6) + (15*x^7 + 30*x^6*y + 56*x^5*y^2 + 74*x^4*y^3 + 74*x^3*y^4 + 56*x^2*y^5 + 30*x*y^6 + 15*y^7) + (22*x^8 + 45*x^7*y + 94*x^6*y^2 + 133*x^5*y^3 + 158*x^4*y^4 + 133*x^3*y^5 + 94*x^2*y^6 + 45*x*y^7 + 22*y^8) + ...

%e such that

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

%e where

%e P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k.

%e SQUARE TABLE.

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

%e 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, ...;

%e 1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, ...;

%e 2, 4, 10, 18, 34, 56, 94, 146, 228, 340, 506, 730, ...;

%e 3, 7, 18, 38, 74, 133, 233, 385, 623, 977, 1501, 2255, ...;

%e 5, 12, 34, 74, 158, 297, 550, 951, 1614, 2627, 4202, 6531, ...;

%e 7, 19, 56, 133, 297, 602, 1166, 2133, 3775, 6437, 10692, ...;

%e 11, 30, 94, 233, 550, 1166, 2382, 4551, 8424, 14953, 25835, ...;

%e 15, 45, 146, 385, 951, 2133, 4551, 9142, 17639, 32680, ...;

%e 22, 67, 228, 623, 1614, 3775, 8424, 17639, 35492, 68356, ...;

%e 30, 97, 340, 977, 2627, 6437, 14953, 32680, 68356, 136936, ...;

%e 42, 139, 506, 1501, 4202, 10692, 25835, 58659, 127443, 264747, ...;

%e 56, 195, 730, 2255, 6531, 17290, 43313, 102149, 229998, 495195, ...;

%e ...

%e TRIANGLE.

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

%e 1;

%e 1, 1;

%e 2, 2, 2;

%e 3, 4, 4, 3;

%e 5, 7, 10, 7, 5;

%e 7, 12, 18, 18, 12, 7;

%e 11, 19, 34, 38, 34, 19, 11;

%e 15, 30, 56, 74, 74, 56, 30, 15;

%e 22, 45, 94, 133, 158, 133, 94, 45, 22;

%e 30, 67, 146, 233, 297, 297, 233, 146, 67, 30;

%e 42, 97, 228, 385, 550, 602, 550, 385, 228, 97, 42;

%e 56, 139, 340, 623, 951, 1166, 1166, 951, 623, 340, 139, 56;

%e 77, 195, 506, 977, 1614, 2133, 2382, 2133, 1614, 977, 506, 195, 77;

%e ...

%p b:= proc(n, i) option remember; expand(`if`(n=0 or i=1,

%p (x+1)^n, b(n, i-1) +(x^i+1)*b(n-i, min(n-i, i))))

%p end:

%p T:= (n, k)-> coeff(b(n+k\$2), x, k):

%p seq(seq(T(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Aug 23 2019

%t b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (x + 1)^n, b[n, i - 1] + (x^i + 1) b[n - i, Min[n - i, i]]]];

%t T[n_, k_] := Coefficient[b[n + k, n + k], x, k];

%t Table[Table[T[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 06 2019, after _Alois P. Heinz_ *)

%o (PARI)

%o {P = 1/prod(n=1,61, (1 - (x^n + y^n) +O(x^61) +O(y^61)) );}

%o {T(n,k) = polcoeff( polcoeff( P,n,x),k,y)}

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

%Y Cf. A322200 (log), A322211 (main diagonal).

%Y Cf. A000041 (row 0 = partitions), A000070 (row 1), A093695(k+2) (row 2).

%Y Antidiagonal sums give A070933.

%Y Cf. A284593.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Nov 30 2018

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Last modified April 11 22:31 EDT 2021. Contains 342895 sequences. (Running on oeis4.)