A038291 Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.

1, 9, 1, 81, 18, 1, 729, 243, 27, 1, 6561, 2916, 486, 36, 1, 59049, 32805, 7290, 810, 45, 1, 531441, 354294, 98415, 14580, 1215, 54, 1, 4782969, 3720087, 1240029, 229635, 25515, 1701, 63, 1, 43046721, 38263752, 14880348, 3306744, 459270, 40824, 2268, 72, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy

Comments

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

Reflected version of A013616. - R. J. Mathar, Dec 19 2008

Triangle of coefficients in expansion of (9 + x)^n, where n is a nonnegative integer. - Zagros Lalo, Jul 21 2018

Examples

			Triangle begins:
  1
  9, 1
  81, 18, 1
  729, 243, 27, 1
  6561, 2916, 486, 36, 1
  59049, 32805, 7290, 810, 45, 1
  531441, 354294, 98415, 14580, 1215, 54, 1
  4782969, 3720087, 1240029, 229635, 25515, 1701, 63, 1
  43046721, 38263752, 14880348, 3306744, 459270, 40824, 2268, 72, 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. A317051, A317052.

Programs

GAP

Flat(List([0..8],i->List([0..i],j->Binomial(i,j)*9^(i-j)*1^j))); # Muniru A Asiru, Jul 21 2018

Maple

for i from 0 to 9 do seq(binomial(i, j)*9^(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, 9 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[(9 + x)^n], x], {n, 0, 8}] // Flatten  (* Zagros Lalo, Jul 22 2018 *)

Formula

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