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A173120 A symmetrical triangle in polynomials of q: q=-4;t(n,m,q)=If[m == 0 || m == n, 1, Binomial[n, m] + q*Binomial[n - 2, m - 1] + q^2*If[n - 4 > 0, Binomial[n - 4, m - 2], 0] + q^3*If[n - 6 > 0, Binomial[n - 6, m - 3], 0] + q^4*If[ n - 8 > 0, Binomial[n - 8, m - 4], 0] + q^5*If[n - 10 > 0, Binomial[n - 10, m - 5], 0]] 0
1, 1, 1, 1, -2, 1, 1, -1, -1, 1, 1, 0, -2, 0, 1, 1, 1, 14, 14, 1, 1, 1, 2, 15, 28, 15, 2, 1, 1, 3, 17, -21, -21, 17, 3, 1, 1, 4, 20, -4, -42, -4, 20, 4, 1, 1, 5, 24, 16, 210, 210, 16, 24, 5, 1, 1, 6, 29, 40, 226, 420, 226, 40, 29, 6, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, 2, 0, 0, 0, 32, 64, 0, 0, 512, 1024,...}.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

q=-4;

t(n,m,q)=If[m == 0 || m == n, 1, Binomial[n, m] +

q*Binomial[n - 2, m - 1] +

q^2*If[n - 4 > 0, Binomial[n - 4, m - 2], 0] +

q^3*If[n - 6 > 0, Binomial[n - 6, m - 3], 0] +

q^4*If[ n - 8 > 0, Binomial[n - 8, m - 4], 0] +

q^5*If[n - 10 > 0, Binomial[n - 10, m - 5], 0]]

EXAMPLE

{1},

{1, 1},

{1, -2, 1},

{1, -1, -1, 1},

{1, 0, -2, 0, 1},

{1, 1, 14, 14, 1, 1},

{1, 2, 15, 28, 15, 2, 1},

{1, 3, 17, -21, -21, 17, 3, 1},

{1, 4, 20, -4, -42, -4, 20, 4, 1},

{1, 5, 24, 16, 210, 210, 16, 24, 5, 1},

{1, 6, 29, 40, 226, 420, 226, 40, 29, 6, 1}

MATHEMATICA

Clear[t, n, m, q];

t[n_, m_, q_] := If[m == 0 || m == n, 1, Binomial[n, m] +

q*Binomial[n - 2, m - 1] +

q^2*If[n - 4 > 0, Binomial[n - 4, m - 2], 0] +

q^3*If[n - 6 > 0, Binomial[n - 6, m - 3], 0] +

q^4*If[ n - 8 > 0, Binomial[n - 8, m - 4], 0] +

q^5*If[n - 10 > 0, Binomial[n - 10, m - 5], 0]];

Table[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 1, 10}];

Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 10}]

CROSSREFS

Sequence in context: A114117 A144435 A182533 * A025920 A037821 A316863

Adjacent sequences:  A173117 A173118 A173119 * A173121 A173122 A173123

KEYWORD

sign,tabl,uned

AUTHOR

Roger L. Bagula, Feb 10 2010

STATUS

approved

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Last modified November 17 10:37 EST 2019. Contains 329225 sequences. (Running on oeis4.)