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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 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 May 17 15:51 EDT 2021. Contains 343980 sequences. (Running on oeis4.)