A038243 Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).

1, 5, 1, 25, 10, 1, 125, 75, 15, 1, 625, 500, 150, 20, 1, 3125, 3125, 1250, 250, 25, 1, 15625, 18750, 9375, 2500, 375, 30, 1, 78125, 109375, 65625, 21875, 4375, 525, 35, 1, 390625, 625000, 437500, 175000, 43750, 7000, 700, 40, 1, 1953125, 3515625, 2812500, 1312500, 393750, 78750, 10500, 900, 45, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy

Comments

Mirror image of A013612. - Zerinvary Lajos, Nov 25 2007

T(i,j) is the number of i-permutations of 6 objects a,b,c,d,e,f, with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007

Triangle of coefficients in expansion of (5+x)^n - N-E. Fahssi, Apr 13 2008

Also the convolution triangle of A000351. - Peter Luschny, Oct 09 2022

Examples

			Triangle begins as:
       1;
       5,      1;
      25,     10,      1;
     125,     75,     15,      1;
     625,    500,    150,     20,     1;
    3125,   3125,   1250,    250,    25,    1;
   15625,  18750,   9375,   2500,   375,   30,   1;
   78125, 109375,  65625,  21875,  4375,  525,  35,  1;
  390625, 625000, 437500, 175000, 43750, 7000, 700, 40, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• B. N. Cyvin, J. Brunvoll, and S. J. Cyvin, Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), 109-121.

Crossrefs

Cf. A000351, A000400 (row sums), A036071, A050982, A053464, A081135, A081143.

Sequences of the form q^(n-k)*binomial(n, k): A007318 (q=1), A038207 (q=2), A027465 (q=3), A038231 (q=4), this sequence (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), A027467 (q=15).

Programs

Magma

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

Maple

for i from 0 to 8 do seq(binomial(i, j)*5^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> 5^(n-1)); # Peter Luschny, Oct 09 2022

Mathematica

With[{q=5}, Table[q^(n-k)*Binomial[n,k], {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, May 12 2021 *)

Sage

flatten([[5^(n-k)*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 12 2021

Formula

See A038207 and A027465 and replace 2 and 3 in analogous formulas with 5. - Tom Copeland, Oct 26 2012