A111808 Left half of trinomial triangle (A027907), triangle read by rows.

1, 1, 1, 1, 2, 3, 1, 3, 6, 7, 1, 4, 10, 16, 19, 1, 5, 15, 30, 45, 51, 1, 6, 21, 50, 90, 126, 141, 1, 7, 28, 77, 161, 266, 357, 393, 1, 8, 36, 112, 266, 504, 784, 1016, 1107, 1, 9, 45, 156, 414, 882, 1554, 2304, 2907, 3139, 1, 10, 55, 210, 615, 1452, 2850, 4740, 6765, 8350

Offset

1,5

Comments

Consider a doubly infinite chessboard with squares labeled (n,k), ranks or rows n in Z, files or columns k in Z (Z denotes ...,-2,-1,0,1,2,... ); number of king-paths of length n from (0,0) to (n,k), 0 <= k <= n, is T(n,n-k). - Harrie Grondijs, May 27 2005. Cf. A026300, A114929, A114972.

Triangle of numbers C^(2)(n-1,k), n>=1, of combinations with repetitions from elements {1,2,...,n} over k, such that every element i, i=1,...,n, appears in a k-combination either 0 or 1 or 2 times (cf. also A213742-A213745). - Vladimir Shevelev and Peter J. C. Moses, Jun 19 2012

References

Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Eric Weisstein's World of Mathematics, Trinomial Triangle

Eric Weisstein's World of Mathematics, Trinomial Coefficient

Formula

(1 + x + x^2)^n = Sum(T(n,k)*x^k: 0<=k<=n) + Sum(T(n,k)*x^(2*n-k): 0<=k<n);

T(n, k) = A027907(n, k) = Sum_{i=0,..,(k/2)} binomial(n, n-k+2*i) * binomial(n-k+2*i, i), 0<=k<=n.

T(n, k) = GegenbauerC(k, -n, -1/2). - Peter Luschny, May 09 2016

Maple

T := (n, k) -> simplify(GegenbauerC(k, -n, -1/2)):
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, May 09 2016

Mathematica

Table[GegenbauerC[k, -n, -1/2], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

Crossrefs

Row sums give A027914; central terms give A027908;

T(n, 0) = 0;

T(n, 1) = n for n>1;

T(n, 2) = A000217(n) for n>1;

T(n, 3) = A005581(n) for n>2;

T(n, 4) = A005712(n) for n>3;

T(n, 5) = A000574(n) for n>4;

T(n, 6) = A005714(n) for n>5;

T(n, 7) = A005715(n) for n>6;

T(n, 8) = A005716(n) for n>7;

T(n, 9) = A064054(n-5) for n>8;

T(n, n-5) = A098470(n) for n>4;

T(n, n-4) = A014533(n-3) for n>3;

T(n, n-3) = A014532(n-2) for n>2;

T(n, n-2) = A014531(n-1) for n>1;

T(n, n-1) = A005717(n) for n>0;

T(n, n) = central terms of A027907 = A002426(n).

Sequence in context: A368158 A176850 A208516 * A247046 A081422 A213742

Adjacent sequences: A111805 A111806 A111807 * A111809 A111810 A111811

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Aug 17 2005

Extensions

Corrected and edited by Johannes W. Meijer, Oct 05 2010

Status

approved