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A152250 Eigentriangle, row sums = A001850, the Delannoy numbers. 2
1, 2, 1, 8, 2, 3, 36, 8, 6, 13, 172, 36, 24, 26, 63, 852, 172, 108, 104, 126, 321, 4324, 852, 516, 468, 504, 642, 1683, 22332, 4324, 2556, 2236, 2268, 2568, 3366, 8989, 116876, 22332, 12972, 11076, 10836, 11556, 13464, 17978, 48639 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums = A001850, the Delannoy numbers: (1, 3, 13, 63, 321,...).

Sum of n-th row terms = rightmost term of next row.

LINKS

Table of n, a(n) for n=0..44.

M. Dziemianczuk, Generalizing Delannoy numbers via counting weighted lattice paths, INTEGERS, 13 (2013), #A54.

M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747, 2014

FORMULA

Triangle read by rows, M*Q. M = an infinite lower triangular matrix with A109980 in every column: (1, 2, 8, 36, 172,...); Q = a matrix with A001850 prefaced with a "1" as the main diagonal: (1, 1, 3, 13, 63, 321,...) and the rest zeros.

EXAMPLE

First few rows of the triangle =

1;

2, 1;

8, 2, 3;

36, 8, 6, 13;

172, 36, 24, 26, 63;

852, 172, 108, 104, 126, 321;

4324, 852, 516, 468, 504, 642, 1683;

22332, 4324, 2556, 2236, 2268, 2568, 3366, 8989;

116876, 22332, 12972, 11076, 10836, 11556, 13464, 17978, 48639;

...

Row 3 = (36, 8, 6, 13) = termwise products of (36, 8, 2, 1) and (1, 1, 3, 13).

MATHEMATICA

nmax = 8;

T[0, 0] = 1;

T[n_, 0] := SeriesCoefficient[1/(x + Sqrt[1 - 6x + x^2]), {x, 0, n}];

T[n_, n_] :=  LegendreP[n - 1, 3];

row[n_] := row[n] = Table[T[m, 0], {m, n, 0, -1}]*Table[T[m, m], {m, 0, n} ];

T[n_, k_] /; 0 < k < n := row[n][[k + 1]];

Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Aug 07 2018 *)

CROSSREFS

Cf. A001850, A109980.

Sequence in context: A317932 A253583 A130562 * A154175 A257777 A011208

Adjacent sequences:  A152247 A152248 A152249 * A152251 A152252 A152253

KEYWORD

eigen,nonn,tabl

AUTHOR

Gary W. Adamson, Nov 30 2008

STATUS

approved

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Last modified October 24 00:04 EDT 2018. Contains 316541 sequences. (Running on oeis4.)