A147716 Triangle of coefficients in expansion of (14 + x)^n.

1, 14, 1, 196, 28, 1, 2744, 588, 42, 1, 38416, 10976, 1176, 56, 1, 537824, 192080, 27440, 1960, 70, 1, 7529536, 3226944, 576240, 54880, 2940, 84, 1, 105413504, 52706752, 11294304, 1344560, 96040, 4116, 98, 1, 1475789056, 843308032, 210827008, 30118144, 2689120, 153664, 5488, 112, 1

Offset: 0

Author

Philippe Deléham, Nov 11 2008

Keywords

• easy
• nonn
• tabl

Comments

Triangle T(n,k), read by rows, given by [14, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Examples

			Triangle begins :
       1;
      14,      1;
     196,     28,     1;
    2744,    588,    42,    1;
   38416,  10976,  1176,   56,  1;
  537824, 192080, 27440, 1960, 70, 1;
		

Links

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

Crossrefs

Cf. A001024, A023531, A130595.

Sequences of the form q^(n-k)*binomial(n, k): A007318 (q=1), A038207 (q=2), A027465 (q=3), A038231 (q=4), A038243 (q=5), A038255 (q=6), A027466 (q=7), A038279 (q=8), A038291 (q=9), A038303 (q=10), A038315 (q=11), A038327 (q=12), A133371 (q=13), this sequence (q=14), A027467 (q=15).

Programs

Magma

[14^(n-k)*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 15 2021

Mathematica

With[{m=8}, CoefficientList[CoefficientList[Series[1/(1-14*x-x*y), {x, 0, m}, {y, 0, m}], x], y]]//Flatten (* Georg Fischer, Feb 17 2020 *)

Sage

flatten([[14^(n-k)*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 15 2021

Formula

T(n,k) = binomial(n,k) * 14^(n-k).

G.f.: 1/(1 - 14*x - x*y). - R. J. Mathar, Aug 12 2015

Sum_{k=0..n} T(n, k) = 15^n = A001024(n). - G. C. Greubel, May 15 2021

Extensions

a(36) corrected by Georg Fischer, Feb 17 2020