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A136431 Hyperfibonacci number array read by antidiagonals. 11
0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 4, 3, 0, 1, 4, 7, 7, 5, 0, 1, 5, 11, 14, 12, 8, 0, 1, 6, 16, 25, 26, 20, 13, 0, 1, 7, 22, 41, 51, 46, 33, 21, 0, 1, 8, 29, 63, 92, 97, 79, 54, 34, 0, 1, 9, 37, 92, 155, 189, 176, 133, 88, 55, 0, 1, 10, 46, 129, 247, 344, 365, 309, 221, 143, 89, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

"In this work, we introduce a symmetric algorithm obtained by the recurrence relation a_{n}^{k}=a_{n-1}^{k}+a_{n}^{k-1}. We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacci- and Lucas numbers. An explicit formulas for hyperharmonic numbers, general generating functions of the Fibonacci- and Lucas numbers are obtained. Besides we define "hyperfibonacci numbers", "hyperlucas numbers". Using these new concepts, some relations between ordinary and incomplete Fibonacci- and Lucas numbers are investigated." [Dil and Mezo]

Main diagonal is A108081. Antidiagonal sums form A027934. - Gerald McGarvey, Oct 01 2008

Seen as triangle read by rows: T(n,0) = 1, T(n,n) = A000045(n) and for 0 < k < n: T(n,k) = T(n-1,k-1) + T(n-1,k). - Reinhard Zumkeller, Jul 16 2013

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

H. Belbachir, A. Belkhir, Combinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers, J. Int. Seq. 17 (2014) # 14.4.3

Ayhan Dil and Istvan Mezo, A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers, arXiv:0803.4388 [math.NT], 2008.

Eric W. Weisstein, Steenrod Algebra

FORMULA

a(k,n) = Apply partial sum operator k times to Fibonacci numbers.

For k > 0 and n > 1, a(k,n) = a(k-1,n) + a(k,n-1). - Gerald McGarvey, Oct 01 2008

EXAMPLE

The array F(n)^{k} begins:

.....|n=0|n=1|.n=2|.n=3|.n=4.|.n=5.|..n=6.|.n=7..|..n=8..|..n=9..|.n=10..|.in.OEIS

k=0..|.0.|.1.|..1.|..2.|...3.|...5.|....8.|...13.|....21.|....34.|....55.|.A000045

k=1..|.0.|.1.|..2.|..4.|...7.|..12.|...20.|...33.|....54.|....88.|...143.|.A000071

k=2..|.0.|.1.|..3.|..7.|..14.|..26.|...46.|...79.|...133.|...221.|...364.|.A001924

k=3..|.0.|.1.|..4.|.11.|..25.|..51.|...97.|..176.|...309.|...530.|...894.|.A014162

k=4..|.0.|.1.|..5.|.16.|..41.|..92.|..189.|..365.|...674.|..1204.|..2098.|.A014166

k=5..|.0.|.1.|..6.|.22.|..63.|.155.|..344.|..709.|..1383.|..2587.|..4685.|.A053739

k=6..|.0.|.1.|..7.|.29.|..92.|.247.|..591.|.1300.|..2683.|..5270.|..9955.|.A053295

k=7..|.0.|.1.|..8.|.37.|.129.|.376.|..967.|.2267.|..4950.|.10220.|.20175.|.A053296

k=8..|.0.|.1.|..9.|.46.|.175.|.551.|.1518.|.3785.|..8735.|.18955.|.39130.|.A053308

k=9..|.0.|.1.|.10.|.56.|.231.|.782.|.2300.|.6085.|.14820.|.33775.|.72905.|.A053309

MAPLE

A136431 := proc(k, n) local x ; coeftayl(x/(1-x-x^2)/(1-x)^k, x=0, n) ; end: for d from 0 to 20 do for n from 0 to d do printf("%d, ", A136431(d-n, n)) ; od: od: # R. J. Mathar, Apr 25 2008

MATHEMATICA

t[n_, k_] := CoefficientList[Series[x/(1 - x - x^2)/(1 - x)^k, {x, 0, n + 1}], x][[n + 1]]; Table[ t[n, k - n], {k, 0, 11}, {n, 0, k}] // Flatten

(* To view the table above *) Table[ t[n, k], {k, 0, 9}, {n, 0, 10}] // TableForm

PROG

(Haskell)

a136431 n k = a136431_tabl !! n !! k

a136431_row n = a136431_tabl !! n

a136431_tabl = map fst $ iterate h ([0], 1) where

   h (row, fib) = (zipWith (+) ([0] ++ row) (row ++ [fib]), last row)

-- Reinhard Zumkeller, Jul 16 2013

CROSSREFS

Cf. A000045, A000071, A001924, A014162, A014166, A053739, A053295, A053296, A053308, A053309, A123736.

Sequence in context: A059259 A124394 A086460 * A182888 A296068 A144064

Adjacent sequences:  A136428 A136429 A136430 * A136432 A136433 A136434

KEYWORD

easy,nonn,tabl,changed

AUTHOR

Jonathan Vos Post, Apr 01 2008

STATUS

approved

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Last modified February 18 20:32 EST 2018. Contains 299330 sequences. (Running on oeis4.)