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A063265 Septinomial (also called heptanomial) coefficient array.

%I A063265 #25 Dec 18 2024 09:22:13
%S A063265 1,1,1,1,1,1,1,1,1,2,3,4,5,6,7,6,5,4,3,2,1,1,3,6,10,15,21,28,33,36,37,
%T A063265 36,33,28,21,15,10,6,3,1,1,4,10,20,35,56,84,116,149,180,206,224,231,
%U A063265 224,206,180,149,116,84,56,35
%N A063265 Septinomial (also called heptanomial) coefficient array.
%C A063265 The sequence of step width of this staircase array is [1,6,6,...], hence the degree sequence for the row polynomials is [0,6,12,18,...]= A008588.
%C A063265 The column sequences (without leading zeros) are for k=0..6 those of the lower triangular array A007318 (Pascal) and for k=7..9: A063267, A063417, A063418. Row sums give A000420 (powers of 7). Central coefficients give A025012.
%D A063265 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.
%H A063265 T. D. Noe, <a href="/A063265/b063265.txt">Rows n = 0..25, flattened</a>
%H A063265 Steven R. Finch, P. Sebah and Z.-Q. Bai, <a href="http://arXiv.org/abs/0802.2654">Odd Entries in Pascal's Trinomial Triangle</a>, arXiv:0802.2654 [math.NT], 2008.
%F A063265 a(n, k)=0 if n=-1 or k<0 or k >= 6*n; a(0, 0)=1; a(n, k)= sum(a(n-1, k-j), j=0..6) else.
%F A063265 G.f. for row n: (sum(x^j, j=0..6))^n.
%F A063265 G.f. for column k: (x^(ceiling(k/6)))*N7(k, x)/(1-x)^(k+1) with the row polynomials of the staircase array A063266(k, m).
%F A063265 T(n,k) = Sum_{i = 0..floor(k/7)} (-1)^i*binomial(n,i)*binomial(n+k-1-7*i,n-1) for n >= 0 and 0 <= k <= 6*n. - _Peter Bala_, Sep 07 2013
%e A063265 Triangle begins:
%e A063265   {1};
%e A063265   {1, 1, 1, 1, 1, 1, 1};
%e A063265   {1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1};
%e A063265   ...
%e A063265 N7(k,x)= 1 for k=0..6, N7(7,x)= 6-15*x+20*x^2-15*x^3+6*x^4-x^5 (from A063266).
%p A063265 #Define the r-nomial coefficients for r = 1, 2, 3, ...
%p A063265 rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
%p A063265 #Display the 7-nomials as a table
%p A063265 r := 7:  rows := 10:
%p A063265 for n from 0 to rows do
%p A063265 seq(rnomial(r,n,k), k = 0..(r-1)*n)
%p A063265 end do;
%p A063265 # _Peter Bala_, Sep 07 2013
%t A063265 Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)^n, x], {n, 0, 25}]] (* _T. D. Noe_, Apr 04 2011 *)
%Y A063265 The q-nomial arrays are for q=2..8: A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171890.
%K A063265 nonn,easy,tabf
%O A063265 0,10
%A A063265 _Wolfdieter Lang_, Jul 24 2001