A027467 Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).

1, 15, 1, 225, 30, 1, 3375, 675, 45, 1, 50625, 13500, 1350, 60, 1, 759375, 253125, 33750, 2250, 75, 1, 11390625, 4556250, 759375, 67500, 3375, 90, 1, 170859375, 79734375, 15946875, 1771875, 118125, 4725, 105, 1, 2562890625, 1366875000, 318937500, 42525000, 3543750, 189000, 6300, 120, 1

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• tabl

Examples

			Triangle begins:
           1;
          15,          1;
         225,         30,         1;
        3375,        675,        45,        1;
       50625,      13500,      1350,       60,       1;
      759375,     253125,     33750,     2250,      75,      1;
    11390625,    4556250,    759375,    67500,    3375,     90,    1;
   170859375,   79734375,  15946875,  1771875,  118125,   4725,  105,   1;
  2562890625, 1366875000, 318937500, 42525000, 3543750, 189000, 6300, 120, 1;
		

Links

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

Crossrefs

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), A147716 (q=14), this sequence (q=15).

Programs

Magma

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

Mathematica

Table[Binomial[n,k]15^(n-k),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Dec 31 2017 *)

Sage

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

Formula

Numerators of lower triangle of (a[i,j])^4 where a[i,j] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.

Sum_{k=0..n} T(n,k)*x^k = (15 + x)^n.

Extensions

Simpler definition from Philippe Deléham, Nov 10 2008