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A355341
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G.f.: A(x) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
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9
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2, 1, -2, 1, -3, 0, 1, -4, 2, 0, 1, -5, 5, 0, 0, 1, -6, 9, -2, 0, 0, 1, -7, 14, -7, 0, 0, 0, 1, -8, 20, -16, 2, 0, 0, 0, 1, -9, 27, -30, 9, 0, 0, 0, 0, 1, -10, 35, -50, 25, -2, 0, 0, 0, 0, 1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0, 1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0, 1, -13, 65, -156, 182, -91, 13, 0, 0, 0, 0, 0, 0, 1, -14, 77, -210, 294, -196, 49, -2, 0, 0, 0, 0, 0, 0, 1, -15, 90, -275, 450, -378, 140, -15, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,1
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n) * x^n is equal to the following expressions; here, C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(1) A(x) = 1/C(x) * Product_{n>=1} (1 + x^n/C(x)) * (1 + x^(n-1)*C(x)) * (1-x^n), by the Jacobi triple product identity.
(2) A(x) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^n.
(3) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * (C(x)^n + 1/C(x)^(n+1)).
(4) A(x) = 1 + Sum_{n>=0} x^(n*(n+1)/2) * ( [y^n] (1 - 2*y*x)/(1-y + x*y^2) ).
(5) A(x) = 1 + Sum_{n>=1} x^(n*(n-1)/2) * Sum_{k=0..n} A244422(n,k) * x^k.
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EXAMPLE
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G.f.: A(x) = 2 + x - 2*x^2 + x^3 - 3*x^4 + x^6 - 4*x^7 + 2*x^8 + x^10 - 5*x^11 + 5*x^12 + x^15 - 6*x^16 + 9*x^17 - 2*x^18 + x^21 - 7*x^22 + 14*x^23 - 7*x^24 + x^28 - 8*x^29 + 20*x^30 - 16*x^31 + 2*x^32 + x^36 - 9*x^37 + 27*x^38 - 30*x^39 + 9*x^40 + x^45 - 10*x^46 + 35*x^47 - 50*x^48 + 25*x^49 - 2*x^50 + ...
such that
A(x) = ... + x^6/C(x)^4 + x^3/C(x)^3 + x/C(x)^2 + 1/C(x) + 1 + x*C(x) + x^3*C(x)^2 + x^6*C(x)^3 + x^10*C(x)^4 + ...
where
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + 16796*x^10 + ... + A000108(n)*x^n + ...
The coefficients of x^k in x^(n*(n+1)/2) * (C(x)^n + 1/C(x)^(n+1)) begin:
n = 0: [2, -1, -1, -2, -5, -14, -42, -132, -429, -1430, -4862, -16796, ...];
n = 1: [0, 2, -1, 1, 3, 9, 28, 90, 297, 1001, 3432, 11934, ...];
n = 2: [0, 0, 0, 2, -1, 5, 13, 39, 123, 401, 1340, 4565, ...];
n = 3: [0, 0, 0, 0, 0, 0, 2, -1, 11, 28, 89, 293, ...];
n = 4: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -1, ...]; ...
forming a table the column sums of which yield this sequence.
The g.f. may also be written as
A(x) = 2 + (-2*x + 1)*x + (-3*x + 1)*x^3 + (2*x^2 - 4*x + 1)*x^6 + (5*x^2 - 5*x + 1)*x^10 + (-2*x^3 + 9*x^2 - 6*x + 1)*x^15 + (-7*x^3 + 14*x^2 - 7*x + 1)*x^21 + (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)*x^28 + (9*x^4 - 30*x^3 + 27*x^2 - 9*x + 1)*x^36 + (-2*x^5 + 25*x^4 - 50*x^3 + 35*x^2 - 10*x + 1)*x^45 + ...
compare to
(1 - 2*y*x)/(1-x + y*x^2) = 1 + (-2*y + 1)*x + (-3*y + 1)*x^2 + (2*y^2 - 4*y + 1)*x^3 + (5*y^2 - 5*y + 1)*x^4 + (-2*y^3 + 9*y^2 - 6*y + 1)*x^5 + (-7*y^3 + 14*y^2 - 7*y + 1)*x^6 + (2*y^4 - 16*y^3 + 20*y^2 - 8*y + 1)*x^7 + (9*y^4 - 30*y^3 + 27*y^2 - 9*y + 1)*x^8 + (-2*y^5 + 25*y^4 - 50*y^3 + 35*y^2 - 10*y + 1)*x^9 + ...
The terms of this sequence may be written as a triangle (see triangle A244422):
2,
1, -2,
1, -3, 0,
1, -4, 2, 0,
1, -5, 5, 0, 0,
1, -6, 9, -2, 0, 0,
1, -7, 14, -7, 0, 0, 0,
1, -8, 20, -16, 2, 0, 0, 0,
1, -9, 27, -30, 9, 0, 0, 0, 0,
1, -10, 35, -50, 25, -2, 0, 0, 0, 0,
1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0,
1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0,
1, -13, 65, -156, 182, -91, 13, 0, 0, 0, 0, 0, 0,
1, -14, 77, -210, 294, -196, 49, -2, 0, 0, 0, 0, 0, 0,
1, -15, 90, -275, 450, -378, 140, -15, 0, 0, 0, 0, 0, 0, 0,
1, -16, 104, -352, 660, -672, 336, -64, 2, 0, 0, 0, 0, 0, 0, 0,
...
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PROG
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(PARI) {a(n) = my(A, C = serreverse(x-x^2 +x^2*O(x^n))/x);
A = sum(m=-n-1, n+1, x^(m*(m+1)/2) * C^m); polcoeff(A, n)}
for(n=0, 70, print1(a(n), ", "))
(PARI) {a(n) = my(A, C = serreverse(x-x^2 +x^2*O(x^n))/x, M = sqrtint(2*n+9));
A = sum(m=0, M, x^(m*(m+1)/2) * (C^m + 1/C^(m+1))); polcoeff(A, n)}
for(n=0, 70, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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