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Revision History for A303558
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
G.f. A(x) satisfies: 0 = [x^(n-1)] (x*A(x))' / (1 + x*A(x)^n)^n for n>1.
(history; published version)
(history; published version)
EXAMPLE
in which the main diagonal is all zeros after the initial term, illustrating the identity: 0 = [x^(n-1)] (x*A(x))' / A(x)^n for n>1 holds since FA(0) = 1.
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COMMENTS
Related identity: 0 = [x^(n-1)] (x*F(x))' / F(x)^n for n>1 holds when F(0) = 1.
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EXAMPLE
RELATED TABLE.
The table of coefficients of x^k in (x*A(x))' / A(x)^n begins:
n=1: [1, 1, 5, 37, 305, 2516, 19205, 126624, 792081, 10741078, ...];
n=2: [1, 0, 2, 20, 186, 1620, 12495, 79302, 474495, 7998088, ...];
n=3: [1, -1, 0, 8, 100, 969, 7680, 46440, 261588, 5988094, ...];
n=4: [1, -2, -1, 0, 40, 510, 4337, 24608, 126090, 4504280, ...];
n=5: [1, -3, -1, -5, 0, 199, 2120, 11037, 46376, 3386810, ...];
n=6: [1, -4, 0, -8, -25, 0, 749, 3514, 5499, 2514128, ...];
n=7: [1, -5, 2, -10, -39, -116, 0, 290, -9640, 1795607, ...];
n=8: [1, -6, 5, -12, -45, -172, -304, 0, -9000, 1165374, ...];
n=9: [1, -7, 9, -15, -45, -186, -301, 1594, 0, 577155, ...];
n=10: [1, -8, 14, -20, -40, -172, -97, 4278, 11976, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating the identity: 0 = [x^(n-1)] (x*A(x))' / A(x)^n for n>1 holds since F(0) = 1.
CROSSREFS
Cf. A303559.
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approved
editing
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approved
COMMENTS
Compare to: 0 = [x^(n-1)] (x*G(x))' / (1 + x*G(x)^k)^n for n>1 holds when G(x) = 1 + x*G(x)^k and k is fixed; the g.f. of this sequence explores the case where k varies directly with n.
STATUS
approved
editing
STATUS
editing
approved
COMMENTS
Compare to: 0 = [x^(n-1)] (x*G(x))' / (1 + x*G(x)^k)^n for n>1 holds when G(x) = 1 + x*G(x)^k and k is fixed; the g.f. of this sequence describes explores the case where k varies directly with n.
STATUS
approved
editing