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

Author

Reinhard Zumkeller, Aug 17 2005

Keywords

• nonn
• tabl

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

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).

Programs

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 *)

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

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

Extensions

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.