A038279 Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.

1, 8, 1, 64, 16, 1, 512, 192, 24, 1, 4096, 2048, 384, 32, 1, 32768, 20480, 5120, 640, 40, 1, 262144, 196608, 61440, 10240, 960, 48, 1, 2097152, 1835008, 688128, 143360, 17920, 1344, 56, 1, 16777216, 16777216, 7340032, 1835008, 286720

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy

Comments

T(i,j) is the number of i-permutations of 9 objects a,b,c,d,e,f,g,h,i with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007

Triangle of coefficients in expansion of (8 + x)^n, where n is a nonnegative integer. - Zagros Lalo, Jul 21 2018

Examples

			1
8, 1
64, 16, 1
512, 192, 24, 1
4096, 2048, 384, 32, 1
32768, 20480, 5120, 640, 40, 1
262144, 196608, 61440, 10240, 960, 48, 1
2097152, 1835008, 688128, 143360, 17920, 1344, 56, 1
16777216, 16777216, 7340032, 1835008, 286720, 28672, 1792, 64, 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. A317028

Programs

GAP

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

Maple

for i from 0 to 8 do seq(binomial(i, j)*8^(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, 8 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[(8 + x)^n], x], {n, 0, 8}] // Flatten  (* Zagros Lalo, Jul 22 2018 *)
Table[CoefficientList[Binomial[i, j] * 8^(i - j) * 1^j, x], {i, 0, 8}, {j,0, i}] // Flatten (* Zagros Lalo, Jul 23 2018 *)

Formula

T(0,0) = 1; T(n,k) = 8 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