OFFSET
0,2
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
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.
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
EXAMPLE
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
387420489, 387420489, 172186884, 44641044, 7440174, 826686, 61236, 2916, 81, 1
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 *)
PROG
(GAP) Flat(List([0..8], i->List([0..i], j->Binomial(i, j)*9^(i-j)*1^j))); # Muniru A Asiru, Jul 21 2018
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved