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A038303 Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.

%I A038303 #36 Jul 28 2018 11:54:40
%S A038303 1,10,1,100,20,1,1000,300,30,1,10000,4000,600,40,1,100000,50000,10000,
%T A038303 1000,50,1,1000000,600000,150000,20000,1500,60,1,10000000,7000000,
%U A038303 2100000,350000,35000,2100,70,1,100000000,80000000,28000000
%N A038303 Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.
%C A038303 T(i,j) is the number of i-permutations of 11 objects a,b,c,d,e,f,g,h,i,j,k, with repetition allowed, containing j a's. - _Zerinvary Lajos_, Dec 21 2007
%C A038303 Triangle T(n,k), read by rows, given by [10,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Dec 15 2009
%C A038303 Triangle of coefficients in expansion of (10 + x)^n, where n is a nonnegative integer. - _Zagros Lalo_, Jul 21 2018
%D A038303 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48
%H A038303 Muniru A Asiru, <a href="/A038303/b038303.txt">Rows n=0..50 of triangle, flattened</a>
%H A038303 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 A038303 Sum_{k=0..n} T(n,k)*x^k = (10+x)^n. - _Philippe Deléham_, Dec 15 2009
%F A038303 G.f.: -1/(-1+10*x+x*y). - _R. J. Mathar_, Aug 11 2015
%F A038303 T(0,0) = 1; T(n,k) = 10 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 A038303 1
%e A038303 10, 1
%e A038303 100, 20, 1
%e A038303 1000, 300, 30, 1
%e A038303 10000, 4000, 600, 40, 1
%e A038303 100000, 50000, 10000, 1000, 50, 1
%e A038303 1000000, 600000, 150000, 20000, 1500, 60, 1
%e A038303 10000000, 7000000, 2100000, 350000, 35000, 2100, 70, 1
%e A038303 100000000, 80000000, 28000000, 5600000, 700000, 56000, 2800, 80, 1
%p A038303 for i from 0 to 8 do seq(binomial(i, j)*10^(i-j), j = 0 .. i) od; # _Zerinvary Lajos_, Dec 21 2007
%t A038303 t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 10 t[n - 1, k] + t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Zagros Lalo_, Jul 21 2018 *)
%t A038303 Table[CoefficientList[ Expand[(10 + x)^n], x], {n, 0, 8}] // Flatten  (* _Zagros Lalo_, Jul 22 2018 *)
%t A038303 Table[CoefficientList[Binomial[i, j] * 10^(i - j) * 1^j, x], {i, 0, 8}, {j, 0, i}] // Flatten (* _Zagros Lalo_, Jul 23 2018 *)
%o A038303 (GAP) Flat(List([0..8],i->List([0..i],j->Binomial(i,j)*10^(i-j)*1^j))); # _Muniru A Asiru_, Jul 21 2018
%Y A038303 Cf. A317054, A317055.
%K A038303 nonn,tabl,easy
%O A038303 0,2
%A A038303 _N. J. A. Sloane_