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A322210
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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.
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7
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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, 15, 30, 56, 74, 74, 56, 30, 15, 22, 45, 94, 133, 158, 133, 94, 45, 22, 30, 67, 146, 233, 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
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OFFSET
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0,4
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COMMENTS
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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
From Wouter Meeussen, Mar 09 2023: (Start)
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.
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)
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LINKS
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Alois P. Heinz, Antidiagonals n = 0..200 (first 61 antidiagonals from Paul D. Hanna)
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FORMULA
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FORMULAS FOR TERMS.
T(n,k) = T(k,n) for n >= 0, k >= 0.
T(n,0) = A000041(n) for n >= 0, where A000041 is the partition numbers.
T(n,1) = A000070(n) for n >= 0, where A000070 is the sum of partitions.
ROW GENERATING FUNCTIONS.
Row 0: 1/( Product_{n>=1} (1 - x^n) ).
Row 1: 1/( (1-x) * Product_{n>=1} (1 - x^n) ).
Row 2: 2/( (1-x) * (1-x^2) * Product_{n>=1} (1 - x^n) ).
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EXAMPLE
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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) + ...
such that
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.
SQUARE TABLE.
The square table of coefficients T(n,k) of x^n*y^k in P(x,y) begins
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, ...;
1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, ...;
2, 4, 10, 18, 34, 56, 94, 146, 228, 340, 506, 730, ...;
3, 7, 18, 38, 74, 133, 233, 385, 623, 977, 1501, 2255, ...;
5, 12, 34, 74, 158, 297, 550, 951, 1614, 2627, 4202, 6531, ...;
7, 19, 56, 133, 297, 602, 1166, 2133, 3775, 6437, 10692, ...;
11, 30, 94, 233, 550, 1166, 2382, 4551, 8424, 14953, 25835, ...;
15, 45, 146, 385, 951, 2133, 4551, 9142, 17639, 32680, ...;
22, 67, 228, 623, 1614, 3775, 8424, 17639, 35492, 68356, ...;
30, 97, 340, 977, 2627, 6437, 14953, 32680, 68356, 136936, ...;
42, 139, 506, 1501, 4202, 10692, 25835, 58659, 127443, 264747, ...;
56, 195, 730, 2255, 6531, 17290, 43313, 102149, 229998, 495195, ...;
...
TRIANGLE.
Alternatively, this sequence may be written as a triangle, starting as
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;
15, 30, 56, 74, 74, 56, 30, 15;
22, 45, 94, 133, 158, 133, 94, 45, 22;
30, 67, 146, 233, 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;
77, 195, 506, 977, 1614, 2133, 2382, 2133, 1614, 977, 506, 195, 77;
...
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MAPLE
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b:= proc(n, i) option remember; expand(`if`(n=0 or i=1,
(x+1)^n, b(n, i-1) +(x^i+1)*b(n-i, min(n-i, i))))
end:
T:= (n, k)-> coeff(b(n+k$2), x, k):
seq(seq(T(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Aug 23 2019
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MATHEMATICA
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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[n_, k_] := Coefficient[b[n + k, n + k], x, k];
Table[Table[T[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)
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PROG
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(PARI)
{P = 1/prod(n=1, 61, (1 - (x^n + y^n) +O(x^61) +O(y^61)) ); }
{T(n, k) = polcoeff( polcoeff( P, n, x), k, y)}
for(n=0, 16, for(k=0, 16, print1( T(n, k), ", ") ); print(""))
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CROSSREFS
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Cf. A322200 (log), A322211 (main diagonal).
Cf. A000041 (row 0 = partitions), A000070 (row 1), A093695(k+2) (row 2).
Antidiagonal sums give A070933.
Cf. A284593.
Cf. A361286
Sequence in context: A074829 A060243 A054225 * A228482 A091822 A358178
Adjacent sequences: A322207 A322208 A322209 * A322211 A322212 A322213
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KEYWORD
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nonn,tabl,changed
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AUTHOR
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Paul D. Hanna, Nov 30 2018
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STATUS
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approved
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