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A136431 Hyperfibonacci square number array a(k,n) = F(n)^(k), read by ascending antidiagonals (k, n >= 0).

%I A136431 #37 Feb 16 2025 08:33:07
%S A136431 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,
%T A136431 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,
%U A136431 37,92,155,189,176,133,88,55,0,1,10,46,129,247,344,365,309,221,143,89,0,1
%N A136431 Hyperfibonacci square number array a(k,n) = F(n)^(k), read by ascending antidiagonals (k, n >= 0).
%C A136431 Main diagonal is A108081. Antidiagonal sums form A027934. - _Gerald McGarvey_, Oct 01 2008
%C A136431 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
%H A136431 Reinhard Zumkeller, <a href="/A136431/b136431.txt">Rows n = 0..125 of triangle, flattened</a>
%H A136431 H. Belbachir, A. Belkhir, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Belbachir/belb2.html">Combinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers</a>, J. Int. Seq. 17 (2014), Article #14.4.3.
%H A136431 Ayhan Dil and Istvan Mezo, <a href="http://arXiv.org/abs/0803.4388">A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers</a>, arXiv:0803.4388 [math.NT], 2008.
%H A136431 Ayhan Dil and Istvan Mezo, <a href="https://doi.org/10.1016/j.amc.2008.10.013">A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers</a>, Applied Mathematics and Computation 206(2) (2008), 942-951.
%H A136431 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SteenrodAlgebra.html">Steenrod Algebra</a>
%F A136431 a(k,n) = Apply partial sum operator k times to Fibonacci numbers.
%F A136431 For k > 0 and n > 1, a(k,n) = a(k-1,n) + a(k,n-1). - _Gerald McGarvey_, Oct 01 2008
%e A136431 The array F(n)^(k) begins:
%e A136431 .....|n=0|n=1|.n=2|.n=3|.n=4.|.n=5.|..n=6.|.n=7..|..n=8..|..n=9..|.n=10..|.in.OEIS
%e A136431 k=0..|.0.|.1.|..1.|..2.|...3.|...5.|....8.|...13.|....21.|....34.|....55.|.A000045
%e A136431 k=1..|.0.|.1.|..2.|..4.|...7.|..12.|...20.|...33.|....54.|....88.|...143.|.A000071
%e A136431 k=2..|.0.|.1.|..3.|..7.|..14.|..26.|...46.|...79.|...133.|...221.|...364.|.A001924
%e A136431 k=3..|.0.|.1.|..4.|.11.|..25.|..51.|...97.|..176.|...309.|...530.|...894.|.A014162
%e A136431 k=4..|.0.|.1.|..5.|.16.|..41.|..92.|..189.|..365.|...674.|..1204.|..2098.|.A014166
%e A136431 k=5..|.0.|.1.|..6.|.22.|..63.|.155.|..344.|..709.|..1383.|..2587.|..4685.|.A053739
%e A136431 k=6..|.0.|.1.|..7.|.29.|..92.|.247.|..591.|.1300.|..2683.|..5270.|..9955.|.A053295
%e A136431 k=7..|.0.|.1.|..8.|.37.|.129.|.376.|..967.|.2267.|..4950.|.10220.|.20175.|.A053296
%e A136431 k=8..|.0.|.1.|..9.|.46.|.175.|.551.|.1518.|.3785.|..8735.|.18955.|.39130.|.A053308
%e A136431 k=9..|.0.|.1.|.10.|.56.|.231.|.782.|.2300.|.6085.|.14820.|.33775.|.72905.|.A053309
%p A136431 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
%t A136431 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
%t A136431 (* To view the table above *) Table[ t[n, k], {k, 0, 9}, {n, 0, 10}] // TableForm
%o A136431 (Haskell)
%o A136431 a136431 n k = a136431_tabl !! n !! k
%o A136431 a136431_row n = a136431_tabl !! n
%o A136431 a136431_tabl = map fst $ iterate h ([0], 1) where
%o A136431    h (row, fib) = (zipWith (+) ([0] ++ row) (row ++ [fib]), last row)
%o A136431 -- _Reinhard Zumkeller_, Jul 16 2013
%Y A136431 Cf. A000045, A000071, A001924, A014162, A014166, A053739, A053295, A053296, A053308, A053309, A123736, A137176.
%K A136431 easy,nonn,tabl
%O A136431 0,9
%A A136431 _Jonathan Vos Post_, Apr 01 2008