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%I #29 Jul 28 2018 11:54:29
%S 1,9,1,81,18,1,729,243,27,1,6561,2916,486,36,1,59049,32805,7290,810,
%T 45,1,531441,354294,98415,14580,1215,54,1,4782969,3720087,1240029,
%U 229635,25515,1701,63,1,43046721,38263752,14880348,3306744,459270,40824,2268,72,1
%N Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.
%C T(i,j) is the number of i-permutations of 10 objects a,b,c,d,e,f,g,h,i,j with repetition allowed, containing j a's. - _Zerinvary Lajos_, Dec 21 2007
%C Reflected version of A013616. - _R. J. Mathar_, Dec 19 2008
%C Triangle of coefficients in expansion of (9 + x)^n, where n is a nonnegative integer. - _Zagros Lalo_, Jul 21 2018
%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48
%H Muniru A Asiru, <a href="/A038291/b038291.txt">Rows n=0..50 of triangle, flattened</a>
%H B. N. Cyvin et al., <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), pp. 109-121.
%F T(0,0) = 1; T(n,k) = 9 T(n-1,k) + T(n-1,k-1) for k = 0..n; T(n,k)=0 for n or k < 0. - _Zagros Lalo_, Jul 21 2018
%e 1
%e 9, 1
%e 81, 18, 1
%e 729, 243, 27, 1
%e 6561, 2916, 486, 36, 1
%e 59049, 32805, 7290, 810, 45, 1
%e 531441, 354294, 98415, 14580, 1215, 54, 1
%e 4782969, 3720087, 1240029, 229635, 25515, 1701, 63, 1
%e 43046721, 38263752, 14880348, 3306744, 459270, 40824, 2268, 72, 1
%e 387420489, 387420489, 172186884, 44641044, 7440174, 826686, 61236, 2916, 81, 1
%p for i from 0 to 9 do seq(binomial(i, j)*9^(i-j), j = 0 .. i) od; # _Zerinvary Lajos_, Dec 21 2007
%t t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 9 t[n - 1, k] + t[n - 1, k - 1]];
%t Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Zagros Lalo_, Jul 21 2018*).
%t Table[CoefficientList[ Expand[(9 + x)^n], x], {n, 0, 8}] // Flatten (* _Zagros Lalo_, Jul 22 2018 *)
%o (GAP) Flat(List([0..8],i->List([0..i],j->Binomial(i,j)*9^(i-j)*1^j))); # _Muniru A Asiru_, Jul 21 2018
%Y Cf. A317051, A317052.
%K nonn,tabl,easy
%O 0,2
%A _N. J. A. Sloane_
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