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A038243
Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
7
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
0,2
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
LINKS
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.
FORMULA
See A038207 and A027465 and replace 2 and 3 in analogous formulas with 5. - Tom Copeland, Oct 26 2012
EXAMPLE
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;
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 *)
PROG
(Magma) [5^(n-k)*Binomial(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 12 2021
(SageMath) flatten([[5^(n-k)*binomial(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 12 2021
CROSSREFS
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).
Sequence in context: A123967 A162259 A077195 * A286231 A218016 A193685
KEYWORD
nonn,tabl,easy
STATUS
approved