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%I #49 Jul 28 2018 11:53:13
%S 1,7,1,49,14,1,343,147,21,1,2401,1372,294,28,1,16807,12005,3430,490,
%T 35,1,117649,100842,36015,6860,735,42,1,823543,823543,352947,84035,
%U 12005,1029,49,1,5764801,6588344,3294172,941192,168070,19208,1372,56,1
%N Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).
%C T(i,j) is the number of i-permutations of 8 objects a,b,c,d,e,f,g,h, with repetition allowed, containing j a's. - _Zerinvary Lajos_, Dec 21 2007
%C Triangle of coefficients in the expansion of (7 + 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 Harvey P. Dale, <a href="/A027466/b027466.txt">Table of n, a(n) for n = 0..5000</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 Cube of lower triangular normalized Binomial matrix.
%F Numerators of lower triangle of (a( i, j ))^3 where a( i, j ) = binomial(i-1, j-1)/2^(i-1) if j <= i, 0 otherwise.
%F T(0,0) = 1; T(n,k) = 7 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 7, 1;
%e 49, 14, 1;
%e 343, 147, 21, 1;
%e 2401, 1372, 294, 28, 1;
%e 16807, 12005, 3430, 490, 35, 1;
%e 117649, 100842, 36015, 6860, 735, 42, 1;
%e 823543, 823543, 352947, 84035, 12005, 1029, 49, 1;
%e 5764801, 6588344, 3294172, 941192, 168070, 19208, 1372, 56, 1;
%p for i from 0 to 8 do seq(binomial(i, j)*7^(i-j), j = 0 .. i) od; # _Zerinvary Lajos_, Dec 21 2007
%t Flatten[Table[Binomial[i,j]7^(i-j),{i,0,10},{j,0,i}]] (* _Harvey P. Dale_, Dec 03 2012 *)
%t t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 7 t[n - 1, k] + t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* _Zagros Lalo_, Jul 21 2018 *).
%t Table[CoefficientList[ Expand[(7 + 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)*7^(i-j)))); # _Muniru A Asiru_, Jul 21 2018
%Y Cf. A007318, A038207.
%Y Cf. A317014
%K nonn,tabl,easy
%O 0,2
%A _Olivier Gérard_
%E Simpler definition from _N. J. A. Sloane_
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