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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
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 [math.CO], 2014.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.
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-François Alcover, Aug 07 2018 *)
CROSSREFS
Sequence in context: A253583 A130562 A363894 * A154175 A257777 A011208
KEYWORD
eigen,nonn,tabl
AUTHOR
Gary W. Adamson, Nov 30 2008
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)