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A176198
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Triangle, read by rows, T(n, k) = f(n,k,q) - f(n,0,q) + 1, where f(n, k, q) = [x^k](p(x,n,q)), p(x, n, q) = (1-x)^(n+1)*Sum_{k >= 0} ( (q*k+1)^n + (q*(k+1)-1)^n )*x^k, and q = 2.
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3
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1, 1, 1, 1, 11, 1, 1, 45, 45, 1, 1, 151, 459, 151, 1, 1, 473, 3363, 3363, 473, 1, 1, 1443, 21085, 47095, 21085, 1443, 1, 1, 4357, 121313, 519445, 519445, 121313, 4357, 1, 1, 13103, 663223, 4970575, 9350027, 4970575, 663223, 13103, 1, 1, 39345, 3512679, 43415943, 138826587, 138826587, 43415943, 3512679, 39345, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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T(n, k) = f(n, k, q) - f(n, 0, q) + 1, where f(n, k, q) = [x^k]( p(x, n, q) ), p(x, n, q) = (1-x)^(n+1) * Sum_{k >= 0} ( (q*k + 1)^n + (q*(k+1) - 1)^n )*x^k, and q = 2.
T(n, k) f(n, k, q) - f(n, 0, q) + 1, where f(n, k, q) = [x^k]( p(x, n, q) ), p(x, n, q) = q^n * (1-x)^(n+1) * ( LerchPhi(x, -n, 1/q) + LerchPhi(x, -n, (q-1)/q) ), and q = 2.
T(n, n-k) = T(n, k).
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, 11, 1;
1, 45, 45, 1;
1, 151, 459, 151, 1;
1, 473, 3363, 3363, 473, 1;
1, 1443, 21085, 47095, 21085, 1443, 1;
1, 4357, 121313, 519445, 519445, 121313, 4357, 1;
1, 13103, 663223, 4970575, 9350027, 4970575, 663223, 13103, 1;
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MATHEMATICA
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m:=13;
p[x_, n_, q_]:= (1-x)^(n+1)*Sum[((q*j+1)^n+(q*(j+1)-1)^n)*x^j, {j, 0, m+ 2}];
f[n_, k_, q_]:= Coefficient[Series[p[x, n, q], {x, 0, m+2}], x, k];
T[n_, k_, q_]:= f[n, k, q] - f[n, 0, q] + 1; (* T = A176198 *)
Table[T[n, k, 2], {n, 0, m}, {k, 0, n}]//Flatten
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PROG
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(Magma)
m:=13;
R<x>:=PowerSeriesRing(Integers(), m+2);
p:= func< x, n, q | (1-x)^(n+1)*(&+[((q*j+1)^n + (q*(j+1)-1)^n)*x^j: j in [0..m+2]]) >;
f:= func< n, k, q | Coefficient(R!( p(x, n, q) ), k) >;
T:= func< n, k, q | f(n, k, q) - f(n, 0, q) + 1 >; // T = A176198
[T(n, k, 2): k in [0..n], n in [0..m]]; // G. C. Greubel, Jun 18 2024
(SageMath)
m=13
def p(x, n, q): return (1-x)^(n+1)*sum(((q*j+1)^n + (q*(j+1)-1)^n)*x^j for j in range(m+3))
def f(n, k, q): return ( p(x, n, q) ).series(x, n+1).list()[k]
def T(n, k, q): return f(n, k, q) - f(n, 0, q) + 1 # T = A176198
flatten([[T(n, k, 2) for k in range(n+1)] for n in (0..m)]) # G. C. Greubel, Jun 18 2024
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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