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A099173 Array T(k,n) read by diagonals: g.f. of k-th row x/(1-2x-(k-1)x^2). 0
0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 1, 2, 5, 8, 5, 0, 1, 2, 6, 12, 16, 6, 0, 1, 2, 7, 16, 29, 32, 7, 0, 1, 2, 8, 20, 44, 70, 64, 8, 0, 1, 2, 9, 24, 61, 120, 169, 128, 9, 0, 1, 2, 10, 28, 80, 182, 328, 408, 256, 10, 0, 1, 2, 11, 32, 101, 256, 547, 896, 985, 512, 11, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Table of n, a(n) for n=0..79.

Ralf Stephan, Prove or disprove. 100 Conjectures from the OEIS, #16, arXiv:math/0409509 [math.CO], 2004.

FORMULA

T(k, n) = Sum{i=0..[n/2], k^i * C(n, 2i+1) }.

Recurrence: T(k, 0)=0, T(k, 1)=1, T(k, n) = 2T(k, n-1) + (k-1)T(k, n-2).

T(k, n) = ((1+sqrt(k))^n - (1-sqrt(k))^n)/(2 sqrt(k)). - Jean-François Alcover, Jan 21 2019

EXAMPLE

0,1,2,3,4,5,6,

0,1,2,4,8,16,32,

0,1,2,5,12,29,70,

0,1,2,6,16,44,120,

0,1,2,7,20,61,182,

0,1,2,8,24,80,256,

MATHEMATICA

T[k_, n_] := Which[k==0, n, n==0, 0, True, ((1+Sqrt[k])^n - (1-Sqrt[k])^n)/ (2 Sqrt[k])]; Table[T[k-n, n] // Simplify, {k, 0, 12}, {n, 0, k}] // Flatten (* Jean-François Alcover, Jan 21 2019 *)

PROG

(PARI) T(k, n)=sum(i=0, n\2, k^i*binomial(n, 2*i+1))

CROSSREFS

Rows 0-12 are A001477, A000079, A000129, A002605, A015518, A063727, A002532, A083099, A015519, A003683, A002534, A083102, A015520, A091914.

Columns 0-4 are A000004, A000012, A009056, A008586.

Sequence in context: A298486 A189768 A262881 * A293377 A159880 A289251

Adjacent sequences:  A099170 A099171 A099172 * A099174 A099175 A099176

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Oct 13 2004

STATUS

approved

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Last modified May 25 11:27 EDT 2020. Contains 334592 sequences. (Running on oeis4.)