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

1, 10, 1, 100, 20, 1, 1000, 300, 30, 1, 10000, 4000, 600, 40, 1, 100000, 50000, 10000, 1000, 50, 1, 1000000, 600000, 150000, 20000, 1500, 60, 1, 10000000, 7000000, 2100000, 350000, 35000, 2100, 70, 1, 100000000, 80000000, 28000000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy

Comments

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

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

Triangle of coefficients in expansion of (10 + x)^n, where n is a nonnegative integer. - Zagros Lalo, Jul 21 2018

Examples

			1
10, 1
100, 20, 1
1000, 300, 30, 1
10000, 4000, 600, 40, 1
100000, 50000, 10000, 1000, 50, 1
1000000, 600000, 150000, 20000, 1500, 60, 1
10000000, 7000000, 2100000, 350000, 35000, 2100, 70, 1
100000000, 80000000, 28000000, 5600000, 700000, 56000, 2800, 80, 1
		

References

• Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48

Links

• Muniru A Asiru, Rows n=0..50 of triangle, flattened
• B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Crossrefs

Cf. A317054, A317055.

Programs

GAP

Flat(List([0..8],i->List([0..i],j->Binomial(i,j)*10^(i-j)*1^j))); # Muniru A Asiru, Jul 21 2018

Maple

for i from 0 to 8 do seq(binomial(i, j)*10^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007

Mathematica

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 *)
Table[CoefficientList[ Expand[(10 + x)^n], x], {n, 0, 8}] // Flatten  (* Zagros Lalo, Jul 22 2018 *)
Table[CoefficientList[Binomial[i, j] * 10^(i - j) * 1^j, x], {i, 0, 8}, {j, 0, i}] // Flatten (* Zagros Lalo, Jul 23 2018 *)

Formula

Sum_{k=0..n} T(n,k)*x^k = (10+x)^n. - Philippe Deléham, Dec 15 2009

G.f.: -1/(-1+10*x+x*y). - R. J. Mathar, Aug 11 2015

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