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%I #56 Mar 28 2023 19:02:58
%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 1: the triangular table 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
%C From _Wouter Meeussen_, Mar 09 2023: (Start)
%C Conjecture 2: the square table T(n,k) is the coefficient of s_lambda in the sum over all partitions lambda |-n and nu |-k of (s_rho/mu) where s_lambda*s_mu = Sum(rho|-n+k; C(rho, lambda, mu) s_rho). Simply stated as: multiply lambda with mu, and, for each term in the result, take the skew Schur function with mu and count how often you get the original lambda back. Sum up over all lambda and mu of the size n and k.
%C Conjecture 3: the triangular table T(n,k) is analogous to conjecture 2, but counting s_lambda in s_(lambda/mu) * s_mu with lambda |- n and mu |- k and 0<=k<=n. (End)
%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.
%Y Cf. A361286
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Nov 30 2018
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