A027466 Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).

1, 7, 1, 49, 14, 1, 343, 147, 21, 1, 2401, 1372, 294, 28, 1, 16807, 12005, 3430, 490, 35, 1, 117649, 100842, 36015, 6860, 735, 42, 1, 823543, 823543, 352947, 84035, 12005, 1029, 49, 1, 5764801, 6588344, 3294172, 941192, 168070, 19208, 1372, 56, 1

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• tabl
• easy

Comments

T(i,j) is the number of i-permutations of 8 objects a,b,c,d,e,f,g,h, with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007

Triangle of coefficients in the expansion of (7 + x)^n, where n is a nonnegative integer. - Zagros Lalo, Jul 21 2018

Examples

			Triangle begins:
        1;
        7,       1;
       49,      14,       1;
      343,     147,      21,      1;
     2401,    1372,     294,     28,      1;
    16807,   12005,    3430,    490,     35,     1;
   117649,  100842,   36015,   6860,    735,    42,    1;
   823543,  823543,  352947,  84035,  12005,  1029,   49,  1;
  5764801, 6588344, 3294172, 941192, 168070, 19208, 1372, 56, 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

• Harvey P. Dale, Table of n, a(n) for n = 0..5000
• 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. A007318, A038207.

Cf. A317014

Programs

GAP

Flat(List([0..8],i->List([0..i],j->Binomial(i,j)*7^(i-j)))); # Muniru A Asiru, Jul 21 2018

Maple

for i from 0 to 8 do seq(binomial(i, j)*7^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007

Mathematica

Flatten[Table[Binomial[i,j]7^(i-j),{i,0,10},{j,0,i}]] (* Harvey P. Dale, Dec 03 2012 *)
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 7 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[(7 + x)^n], x], {n, 0, 8}] // Flatten  (* Zagros Lalo, Jul 22 2018 *)

Formula

Cube of lower triangular normalized Binomial matrix.

Numerators of lower triangle of (a( i, j ))^3 where a( i, j ) = binomial(i-1, j-1)/2^(i-1) if j <= i, 0 otherwise.

T(0,0) = 1; T(n,k) = 7*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

Extensions

Simpler definition from N. J. A. Sloane