

A273975


Threedimensional array written by antidiagonals in k,n: T(k,n,h) with k >= 1, n >= 0, 0 <= h <= n*(k1) is the coefficient of x^h in the polynomial (1 + x + ... + x^(k1))^n = ((x^k1)/(x1))^n.


6



1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 6, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10
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OFFSET

1,13


COMMENTS

Equivalently, T(k,n,h) is the number of ordered sets of n nonnegative integers < k with the sum equal to h.
T(k,n,h) is the number of possible ways of randomly selecting h cards from k1 sets, each with n different playing cards. It is also the number of lattice paths from (0,0) to (n,h) using steps (1,0), (1,1), (1,2), ..., (1,k1).
Shallow diagonal sums of each triangle with fixed k give the kbonacci numbers. (End)
T(k,n,h) is the number of ndimensional grid points of a k X k X ... X k grid, which are lying in the (n1)dimensional hyperplane which is at an L1 distance of h from one of the grid's corners, and normal to the corresponding main diagonal of the grid.  Eitan Y. Levine, Apr 23 2023


LINKS

Florentin Smarandache, KNomial Coefficients, arXiv:math/0612062 [math.GM], 2006 (originally published in French in: F. Smarandache, Généralisations et Généralités, Ed. Nouvelle, 1984, pp. 2426).


FORMULA

T(k,n,h) = Sum_{i = 0..floor(h/k)} (1)^i*binomial(n,i)*binomial(n+h1k*i,n1). [Corrected by Eitan Y. Levine, Apr 23 2023]
(T(k,n,h))_{h=0..n*(k1)} = f(f(...f(g(P))...)), where:
(x_i)_{i=0..m} denotes a tuple (in particular, the LHS contains the values for 0 <= h <= n*(k1)),
f repeats n times,
f((x_i)_{i=0..m}) = (Sum_{j=0..i} x_j)_{i=0..m}) is the cumulative sum function,
g((x_i)_{i=0..m}) = (x_(i/k) if ki, otherwise 0)_{i=0..m*k} is adding k1 zeros between adjacent elements,
and P=((1)^i*binomial(n,i))_{i=0..n} is the nth row of Pascal's triangle, with alternating signs. (End)
Recurrence relations, the first follows from the sequence's defining polynomial as mentioned in the Smarandache link:
T(k,n+1,h) = Sum_{i = 0..s1} T(k,n,hi)
T(k+1,n,h) = Sum_{i = 0..n} binomial(n,i)*T(k,ni,hi*k) (End)


EXAMPLE

For first few k and for first few n, the rows with h = 0..n*(k1) are given:
k=1: 1; 1; 1; 1; 1; ...
k=2: 1; 1, 1; 1, 2, 1; 1, 3, 3, 1; 1, 4, 6, 4, 1; ...
k=3: 1; 1, 1, 1; 1, 2, 3, 2, 1; 1, 3, 6, 7, 6, 3, 1; ...
k=4: 1; 1, 1, 1, 1; 1, 2, 3, 4, 3, 2, 1; ...
For example, (1 + x + x^2)^3 = 1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6, hence T(3,3,2) = T(3,3,4) = 6.
Example for the repeated cumulative sum formula, for (k,n)=(3,3) (each line is the cumulative sum of the previous line, and the first line is the padded, alternating 3rd row from Pascal's triangle):
1 0 0 3 0 0 3 0 0 1
1 1 1 2 2 2 1 1 1
1 2 3 1 1 3 2 1
1 3 6 7 6 3 1
which is T(3,3,h). (End)


MATHEMATICA

a = Table[CoefficientList[Sum[x^(h1), {h, k}]^n, x], {k, 10}, {n, 0, 9}];
Flatten@Table[a[[sn, n+1]], {s, 10}, {n, 0, s1}]
(* alternate program *)
row[k_, n_] := Nest[Accumulate, Upsample[Table[((1)^j)*Binomial[n, j], {j, 0, n}], k], n][[;; n*(k1)+1]] (* Eitan Y. Levine, Apr 23 2023 *)


CROSSREFS

Central nnomial coefficients for n=1..9, i.e., sequences with h=floor(n*(k1)/2) and fixed n: A000012, A000984 (A001405), A002426, A005721 (A005190), A005191, A063419 (A018901), A025012, (A025013), A025014, A174061 (A025015), A201549, (A225779), A201550. Arrays: A201552, A077042, see also cfs. therein.


KEYWORD

nonn,tabf,easy


AUTHOR



STATUS

approved



