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A356501
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Coefficients T(n,k) of x^(4*n+1-k)*y^k in A(x,y) for n >= 0, k = 0..3*n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.
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3
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1, 1, 0, 3, 6, 4, 1, 0, 9, 54, 120, 135, 84, 28, 4, 0, 22, 294, 1360, 3250, 4662, 4284, 2568, 981, 219, 22, 0, 51, 1260, 10120, 41405, 103020, 170324, 196172, 160965, 94390, 38896, 10764, 1807, 140, 0, 108, 4590, 58380, 368145, 1404102, 3587696, 6515712, 8715465, 8763645, 6684744, 3863496, 1670942, 525980, 114240, 15368, 969
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OFFSET
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0,4
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COMMENTS
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Alternating row sums equals zero for all rows.
Rightmost border equals [x^n*y^(3*n+1)] A(x,y) = A002293(n) = binomial(4*n, n)/(3*n + 1).
This triangle may be formed from the nonzero antidiagonals of triangle A356500; see main entry A356500 for further formulas for the coefficients in g.f. A(x,y).
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LINKS
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FORMULA
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G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..3*n+1} T(n,k) * x^n * y^k satisfies:
(1) y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n+1)^2).
(2) y = A(x,y) * Product_{n>=1} (1 - x^(2*n)*A(x,y)^(2*n)) * (1 - x^(2*n-1)*A(x,y)^(2*n+1)) * (1 - x^(2*n-1)*A(x,y)^(2*n-3)), by the Jacobi triple product identity.
(3) y = (-x) * Product_{n>=1} (1 - x^(2*n)*A(x,y)^(2*n)) * (1 - x^(2*n+1)*A(x,y)^(2*n-1)) * (1 - x^(2*n-3)*A(x,y)^(2*n-1)), by the Jacobi triple product identity.
(4) y = A(x, F(x,y)) where F(x,y) = Sum_{n=-oo..+oo} (-x)^(n^2) * y^((n-1)^2).
(5) 1 = A(x, theta_4(x)) where theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2) is a Jacobi theta function.
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EXAMPLE
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G.f.: A(x,y) = y + x*(1 + y^4) + x^2*(4*y^3 + 4*y^7) + x^3*(6*y^2 + 28*y^6 + 22*y^10) + x^4*(3*y + 84*y^5 + 219*y^9 + 140*y^13) + x^5*(135*y^4 + 981*y^8 + 1807*y^12 + 969*y^16) + x^6*(120*y^3 + 2568*y^7 + 10764*y^11 + 15368*y^15 + 7084*y^19) + x^7*(54*y^2 + 4284*y^6 + 38896*y^10 + 114240*y^14 + 133266*y^18 + 53820*y^22) + x^8*(9*y + 4662*y^5 + 94390*y^9 + 525980*y^13 + 1187433*y^17 + 1171390*y^21 + 420732*y^25) + x^9*(3250*y^4 + 160965*y^8 + 1670942*y^12 + 6640711*y^16 + 12167001*y^20 + 10399545*y^24 + 3362260*y^28) + ...
such that A = A(x,y) satisfies
y = ... + x^16*A^25 - x^9*A^16 + x^4*A^9 - x*A^4 + A - x + x^4*A - x^9*A^4 + x^16*A^9 - x^25*A^16 +- ... + (-x)^(n^2) * A(x,y)^((n-1)^2) + ...
This triangle of coefficients of x^(4*n+1-k)*y^k in A(x,y) for n >= 0, k = 0..3*n+1, begins:
n = 0: [1, 1];
n = 1: [0, 3, 6, 4, 1];
n = 2: [0, 9, 54, 120, 135, 84, 28, 4];
n = 3: [0, 22, 294, 1360, 3250, 4662, 4284, 2568, 981, 219, 22];
n = 4: [0, 51, 1260, 10120, 41405, 103020, 170324, 196172, 160965, 94390, 38896, 10764, 1807, 140];
n = 5: [0, 108, 4590, 58380, 368145, 1404102, 3587696, 6515712, 8715465, 8763645, 6684744, 3863496, 1670942, 525980, 114240, 15368, 969];
n = 6: [0, 221, 14952, 282948, 2578147, 14039250, 51126740, 133101836, 258436719, 384735141, 446971668, 409367712, 296679006, 169713208, 75904032, 26050408, 6640711, 1187433, 133266, 7084];
...
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PROG
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(PARI) {T(n, k) = my(A=[y], M); for(i=1, 4*n+1, A = concat(A, 0); M = ceil(sqrt(4*n+1));
A[#A] = -polcoeff( sum(m=-M, M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); polcoeff(A[4*n+2-k], k, y)}
for(n=0, 7, for(k=0, 3*n+1, print1(T(n, k), ", ")); print(""))
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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