A147566 Triangle, T(n, k) = coefficients [x^k]( p(x,n) ), where p(x, n) = (x+1)^n for n < 2, otherwise (x+1)^n + x*((1+x)^(n-2) + 2^(n-2)*(1-x)^(n-1)*LerchPhi(x, 2-n, 1/2)), read by rows.

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 6, 14, 6, 1, 1, 7, 36, 36, 7, 1, 1, 8, 95, 256, 95, 8, 1, 1, 9, 263, 1727, 1727, 263, 9, 1, 1, 10, 756, 10614, 23638, 10614, 756, 10, 1, 1, 11, 2222, 60762, 259884, 259884, 60762, 2222, 11, 1, 1, 12, 6605, 331760, 2485554, 4675336, 2485554, 331760, 6605, 12, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = coefficients [x^k]( p(x,n) ), where p(x, n) = (x+1)^n for n < 2, otherwise (x+1)^n + x*((1+x)^(n-2) + 2^(n-2)*(1-x)^(n-1)*LerchPhi(x, 2-n, 1/2)).

T(n, n-k) = T(n, k).

T(n, 1) = A000027(n+2) - 2*[n=1]. - G. C. Greubel, Oct 26 2022

Example

Triangle of coefficients begins as:
  1;
  1,   1;
  1,   4,    1;
  1,   5,    5,     1;
  1,   6,   14,     6,      1;
  1,   7,   36,    36,      7,      1;
  1,   8,   95,   256,     95,      8,     1;
  1,   9,  263,  1727,   1727,    263,     9,    1;
  1,  10,  756, 10614,  23638,  10614,   756,   10,    1;
  1,  11, 2222, 60762, 259884, 259884, 60762, 2222,   11,   1;

Mathematica

p[x_, n_]:= If[n>=2, (x+1)^n + x*((1+x)^(n-2) + 2^(n-2)*(1-x)^(n-1)*LerchPhi[x, 2-n, 1/2]), (x+1)^n];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten

Prog

(Magma) // as a triangle
LerchPhi:= func< x, n, q | (&+[x^k/(k+q)^n: k in [0..100]]) >;
p:= func< n, x | n lt 2 select (x+1)^n else (x+1)^n + x*(1+x)^(n-2) + 2^(n-2)*x*(1-x)^(n-1)*LerchPhi(x, 2-n, 1/2) >;
R<x>:=PowerSeriesRing(Integers(), 30);
[Coefficients(R!( p(n, x) )): n in [0..12]]; // G. C. Greubel, Oct 26 2022
(SageMath)
def LerchPhi(x, n, q): return sum( x^k/(k+q)^n for k in range(100))
def p(n, x):
    if (n<2): return (x+1)^n
    else: return (x+1)^n + x*(1+x)^(n-2) + 2^(n-2)*x*(1-x)^(n-1)*LerchPhi(x, 2-n, 1/2)
flatten([[( p(n, x) ).series(x, n+1).list()[k] for k in range(n+1)] for n in (0..12)]) # G. C. Greubel, Oct 26 2022

Crossrefs

Cf. A000027, A147565.

Sequence in context: A028275 A173118 A147289 * A204621 A146770 A143334

Adjacent sequences: A147563 A147564 A147565 * A147567 A147568 A147569

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 07 2008

Extensions

Edited by G. C. Greubel, Oct 26 2022

Status

approved