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%I #27 Oct 09 2022 05:03:25
%S 1,5,1,25,10,1,125,75,15,1,625,500,150,20,1,3125,3125,1250,250,25,1,
%T 15625,18750,9375,2500,375,30,1,78125,109375,65625,21875,4375,525,35,
%U 1,390625,625000,437500,175000,43750,7000,700,40,1,1953125,3515625,2812500,1312500,393750,78750,10500,900,45,1
%N Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
%C Mirror image of A013612. - _Zerinvary Lajos_, Nov 25 2007
%C 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
%C Triangle of coefficients in expansion of (5+x)^n - _N-E. Fahssi_, Apr 13 2008
%C Also the convolution triangle of A000351. - _Peter Luschny_, Oct 09 2022
%H G. C. Greubel, <a href="/A038243/b038243.txt">Rows n = 0..50 of the triangle, flattened</a>
%H B. N. Cyvin, J. Brunvoll, and S. J. Cyvin, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match34/match34_109-121.pdf">Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons</a>, Match, No. 34 (Oct 1996), 109-121.
%F See A038207 and A027465 and replace 2 and 3 in analogous formulas with 5. - _Tom Copeland_, Oct 26 2012
%e Triangle begins as:
%e 1;
%e 5, 1;
%e 25, 10, 1;
%e 125, 75, 15, 1;
%e 625, 500, 150, 20, 1;
%e 3125, 3125, 1250, 250, 25, 1;
%e 15625, 18750, 9375, 2500, 375, 30, 1;
%e 78125, 109375, 65625, 21875, 4375, 525, 35, 1;
%e 390625, 625000, 437500, 175000, 43750, 7000, 700, 40, 1;
%p for i from 0 to 8 do seq(binomial(i, j)*5^(i-j), j = 0 .. i) od; # _Zerinvary Lajos_, Dec 21 2007
%p # Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
%p PMatrix(10, n -> 5^(n-1)); # _Peter Luschny_, Oct 09 2022
%t With[{q=5}, Table[q^(n-k)*Binomial[n,k], {n,0,12}, {k,0,n}]//Flatten] (* _G. C. Greubel_, May 12 2021 *)
%o (Magma) [5^(n-k)*Binomial(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 12 2021
%o (Sage) flatten([[5^(n-k)*binomial(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 12 2021
%Y Cf. A000351, A000400 (row sums), A036071, A050982, A053464, A081135, A081143.
%Y 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).
%K nonn,tabl,easy
%O 0,2
%A _N. J. A. Sloane_
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