A173122 Irregular triangle T(n) = coefficients of Sum_{k=0..n} t(n,k,q) for powers of q, where t(n,k,q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with t(n,0,q) = t(n,n,q) = 1, read by rows.

1, 2, 4, 1, 8, 2, 16, 4, 32, 8, 2, 64, 16, 4, 128, 32, 8, 2, 256, 64, 16, 4, 512, 128, 32, 8, 2, 1024, 256, 64, 16, 4, 2048, 512, 128, 32, 8, 2, 4096, 1024, 256, 64, 16, 4, 8192, 2048, 512, 128, 32, 8, 16384, 4096, 1024, 256, 64, 16, 32768, 8192, 2048, 512, 128, 32, 65536, 16384, 4096, 1024, 256, 64

Offset

0,2

Links

G. C. Greubel, Rows n = 0..100 of the irregular triangle, flattened

Formula

T(n) = coefficients of Sum_{k=0..n} t(n,k,q) for powers of q, where t(n,k,q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with t(n,0,q) = t(n,n,q) = 1.

Example

Irregular triangle begins as:
     1;
     2;
     4,   1;
     8,   2;
    16,   4;
    32,   8,  2;
    64,  16,  4;
   128,  32,  8,  2;
   256,  64, 16,  4;
   512, 128, 32,  8, 2;
  1024, 256, 64, 16, 4;

Mathematica

t[n_, k_, q_]:= If[k==0 || k==n, 1, q*Boole[n==2] + Sum[q^j*Binomial[n-2*j, k-j] *Boole[n>2*j], {j, 0, 5}]];
T[n_]:= CoefficientList[Series[Sum[t[n, k, q], {k, 0, n}], {q, 0, n}], q];
Table[T[n], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Apr 29 2021 *)

Prog

(Sage)
@CachedFunction
def t(n, k, x): return 1 if (k==0 or k==n) else x*bool(n==2) + sum( x^j*binomial(n-2*j, k-j)*bool(n>2*j) for j in (0..5) )
def s(n, x): return sum( t(n, k, x) for k in (0..n) )
flatten([taylor(s(n, x), x, 0, n).list() for n in (0..12)]) # G. C. Greubel, Apr 29 2021

Crossrefs

Cf. A007318, A072405, A173117, A173118, A173119, A173120.

Sequence in context: A302192 A087060 A248112 * A232723 A275486 A065278

Adjacent sequences: A173119 A173120 A173121 * A173123 A173124 A173125

Keyword

nonn,tabf,easy,less

Author

Roger L. Bagula, Feb 10 2010

Extensions

More terms and edited by G. C. Greubel, Apr 29 2021

Status

approved