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A273975 Three-dimensional array written by antidiagonals in k,n: T(k,n,h) with k>=1, n>=0, 0<=h<=n*(k-1) is the coefficient of x^h in the polynomial (1+x+...+x^(k-1))^n = ((x^k-1)/(x-1))^n. 5
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 (list; graph; refs; listen; history; text; internal format)
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.

From Juan Pablo Herrera P., Nov 21 2016: (Start)

T(k,n,h) is the number of possible ways of randomly selecting h cards from k-1 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,k-1).

Shallow diagonal sums of each triangle with fixed k give the k-bonacci numbers. (End)

LINKS

Andrey Zabolotskiy, slices with k+n = 1..20, flattened

Florentin Smarandache, K-Nomial Coefficients, arXiv:math/0612062 [math.GM], 2006 (originally published in French in: F. Smarandache, Généralisations et Généralités, Ed. Nouvelle, 1984, pp. 24-26).

FORMULA

T(k,n,h) = Sum_{i = 0..floor(h/k)} (-1)^i*binomial(n,i)*binomial(n+h-1-k*i,k-1).

EXAMPLE

For first few k and for first few n, the rows with h = 0..n*(k-1) 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.

MATHEMATICA

a = Table[CoefficientList[Sum[x^(h-1), {h, k}]^n, x], {k, 10}, {n, 0, 9}];

Flatten[Table[a[[s-n, n+1]], {s, 10}, {n, 0, s-1}]

CROSSREFS

k-nomial arrays for fixed k=1..10: A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.

Arrays for fixed n=0..6: A000012, A000012, A004737, A109439, A277949, A277950, A277951.

Central n-nomial coefficients for n=1..9: A000012, A000984 (A001405), A002426, A005721 (A005190), A005191, A063419 (A018901), A025012, (A025013), A025014, A174061 (A025015), A201549, (A225779), A201550. Arrays: A201552, A077042, see also cfs. therein.

Triangle n=k-1: A181567. Triangle n=k: A163181.

Sequence in context: A031276 A261794 A098744 * A025429 A076250 A231425

Adjacent sequences:  A273972 A273973 A273974 * A273976 A273977 A273978

KEYWORD

nonn,tabf,easy

AUTHOR

Andrey Zabolotskiy, Nov 10 2016

STATUS

approved

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Last modified February 22 05:58 EST 2018. Contains 299430 sequences. (Running on oeis4.)