cp's OEIS Frontend

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.

%I A322210 #63 Apr 13 2025 05:12:14
%S A322210 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 A322210 15,30,56,74,74,56,30,15,22,45,94,133,158,133,94,45,22,30,67,146,233,
%U A322210 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 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.
%C A322210 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 A322210 From _Wouter Meeussen_, Mar 09 2023: (Start)
%C A322210 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 A322210 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 A322210 Alois P. Heinz, <a href="/A322210/b322210.txt">Antidiagonals n = 0..200</a> (first 61 antidiagonals from Paul D. Hanna)
%F A322210 FORMULAS FOR TERMS.
%F A322210 T(n,k) = T(k,n) for n >= 0, k >= 0.
%F A322210 T(n,0) = A000041(n) for n >= 0, where A000041 is the partition numbers.
%F A322210 T(n,1) = A000070(n) for n >= 0, where A000070 is the sum of partitions.
%F A322210 ROW GENERATING FUNCTIONS.
%F A322210 Row 0: 1/( Product_{n>=1} (1 - x^n) ).
%F A322210 Row 1: 1/( (1-x) * Product_{n>=1} (1 - x^n) ).
%F A322210 Row 2: 2/( (1-x) * (1-x^2) * Product_{n>=1} (1 - x^n) ).
%e A322210 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 A322210 such that
%e A322210 P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)),
%e A322210 where
%e A322210 P(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k.
%e A322210 SQUARE TABLE.
%e A322210 The square table of coefficients T(n,k) of x^n*y^k in P(x,y) begins
%e A322210    1,   1,   2,    3,    5,     7,    11,     15,     22,     30, ...
%e A322210    1,   2,   4,    7,   12,    19,    30,     45,     67,     97, ...
%e A322210    2,   4,  10,   18,   34,    56,    94,    146,    228,    340, ...
%e A322210    3,   7,  18,   38,   74,   133,   233,    385,    623,    977, ...
%e A322210    5,  12,  34,   74,  158,   297,   550,    951,   1614,   2627, ...
%e A322210    7,  19,  56,  133,  297,   602,  1166,   2133,   3775,   6437, ...
%e A322210   11,  30,  94,  233,  550,  1166,  2382,   4551,   8424,  14953, ...
%e A322210   15,  45, 146,  385,  951,  2133,  4551,   9142,  17639,  32680, ...
%e A322210   22,  67, 228,  623, 1614,  3775,  8424,  17639,  35492,  68356, ...
%e A322210   30,  97, 340,  977, 2627,  6437, 14953,  32680,  68356, 136936, ...
%e A322210   42, 139, 506, 1501, 4202, 10692, 25835,  58659, 127443, 264747, ...
%e A322210   56, 195, 730, 2255, 6531, 17290, 43313, 102149, 229998, 495195, ...
%e A322210   ...
%e A322210 TRIANGLE.
%e A322210 Alternatively, this sequence may be written as a triangle, starting as
%e A322210    1;
%e A322210    1,   1;
%e A322210    2,   2,   2;
%e A322210    3,   4,   4,   3;
%e A322210    5,   7,  10,   7,    5;
%e A322210    7,  12,  18,  18,   12,    7;
%e A322210   11,  19,  34,  38,   34,   19,   11;
%e A322210   15,  30,  56,  74,   74,   56,   30,   15;
%e A322210   22,  45,  94, 133,  158,  133,   94,   45,   22;
%e A322210   30,  67, 146, 233,  297,  297,  233,  146,   67,  30;
%e A322210   42,  97, 228, 385,  550,  602,  550,  385,  228,  97,  42;
%e A322210   56, 139, 340, 623,  951, 1166, 1166,  951,  623, 340, 139,  56;
%e A322210   77, 195, 506, 977, 1614, 2133, 2382, 2133, 1614, 977, 506, 195, 77;
%e A322210   ...
%p A322210 b:= proc(n, i) option remember; expand(`if`(n=0 or i=1,
%p A322210       (x+1)^n, b(n, i-1) +(x^i+1)*b(n-i, min(n-i, i))))
%p A322210     end:
%p A322210 T:= (n, k)-> coeff(b(n+k$2), x, k):
%p A322210 seq(seq(T(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Aug 23 2019
%t A322210 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 A322210 T[n_, k_] := Coefficient[b[n + k, n + k], x, k];
%t A322210 Table[Table[T[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 06 2019, after _Alois P. Heinz_ *)
%o A322210 (PARI)
%o A322210 {P = 1/prod(n=1,61, (1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
%o A322210 {T(n,k) = polcoeff( polcoeff( P,n,x),k,y)}
%o A322210 for(n=0,16, for(k=0,16, print1( T(n,k),", ") );print(""))
%Y A322210 Cf. A322200 (log).
%Y A322210 Cf. A000041 (row 0 = partitions), A000070 (row 1), A093695(k+2) (row 2).
%Y A322210 Main diagonal gives A322211.
%Y A322210 Antidiagonal sums give A070933.
%Y A322210 Cf. A284593.
%Y A322210 Cf. A361286.
%K A322210 nonn,tabl
%O A322210 0,4
%A A322210 _Paul D. Hanna_, Nov 30 2018