A108081 -id:A108081 - cp's OEIS frontend

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

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

Offset: 0

Jonathan Vos Post, Apr 01 2008

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

			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

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), Article #14.4.3.
Ayhan Dil and Istvan Mezo, A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers, arXiv:0803.4388 [math.NT], 2008.
Ayhan Dil and Istvan Mezo, A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers, Applied Mathematics and Computation 206(2) (2008), 942-951.
Eric Weisstein's World of Mathematics, Steenrod Algebra

Cf. A000045, A000071, A001924, A014162, A014166, A053739, A053295, A053296, A053308, A053309, A123736, A137176.

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

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[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

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

A371742 a(n) = Sum_{k=0..floor(n/2)} binomial(3n-k,n-2k).

Original entry on oeis.org

1, 3, 16, 92, 551, 3380, 21065, 132771, 843944, 5399802, 34731776, 224361283, 1454557294, 9458829681, 61670895633, 403003997300, 2638776935215, 17308508054848, 113709379928689, 748069400432262, 4927608724973776, 32495826854732633, 214521754579553129

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A108081, A371743, A371744.
Cf. A066380, A371754.
Cf. A183160.

a(n) = sum(k=0, n\2, binomial(3*n-k, n-2*k));

a(n) = [x^n] 1/((1-x-x^2) * (1-x)^(2*n)).

a(n) ~ 3^(3*n + 3/2) / (5 * sqrt(Pi*n) * 2^(2*n)). - Vaclav Kotesovec, Apr 05 2024

A159965 Riordan array (1/sqrt(1-4x), (1-2x-(1-3x)c(x))/(x*sqrt(1-4x))), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 20, 21, 8, 1, 70, 84, 45, 11, 1, 252, 330, 220, 78, 14, 1, 924, 1287, 1001, 455, 120, 17, 1, 3432, 5005, 4368, 2380, 816, 171, 20, 1, 12870, 19448, 18564, 11628, 4845, 1330, 231, 23, 1, 48620, 75582, 77520, 54264, 26334, 8855, 2024, 300, 26, 1

Offset: 0

Paul Barry, Apr 28 2009

Product of A007318 and A114422. Product of A007318^2 and A116382. Row sums are A108080.
Diagonal sums are A108081.
Riordan array (1/sqrt(1 - 4*x), x*c(x)^3) obtained from A092392 by taking every third column starting from column 0; x*c(x)^3 is the o.g.f. for A000245. - Peter Bala, Nov 24 2015

			Triangle begins
1,
2, 1,
6, 5, 1,
20, 21, 8, 1,
70, 84, 45, 11, 1,
252, 330, 220, 78, 14, 1,
924, 1287, 1001, 455, 120, 17, 1,
3432, 5005, 4368, 2380, 816, 171, 20, 1

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

Cf. A000245, A007318, A092392, A108080, A108081, A114422, A116382.

/* As triangle */ [[Binomial(2*n+k, n+2*k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 27 2015

Number triangle T(n,k) = Sum_{j = 0..n} binomial(n+k,j-k)*binomialC(n,j).

T(n,k) = binomial(2*n + k, n + 2*k). - Peter Bala, Nov 24 2015

A371743 a(n) = Sum_{k=0..floor(n/2)} binomial(4n-k,n-2k).

Original entry on oeis.org

1, 4, 29, 231, 1926, 16491, 143683, 1267395, 11282393, 101151544, 912011633, 8260998772, 75115815749, 685232639419, 6268299350776, 57478389714473, 528167137069958, 4862304525663579, 44836026545219765, 414048025058547788, 3828677665694353049

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A108081, A371742, A371744.

a(n) = sum(k=0, n\2, binomial(4*n-k, n-2*k));

a(n) = [x^n] 1/((1-x-x^2) * (1-x)^(3*n)).

a(n) ~ 2^(8*n + 5/2) / (11 * sqrt(Pi*n) * 3^(3*n - 1/2)). - Vaclav Kotesovec, Apr 05 2024

A371744 a(n) = Sum_{k=0..floor(n/2)} binomial(5n-k,n-2k).

Original entry on oeis.org

1, 5, 46, 469, 5017, 55177, 617905, 7008264, 80241790, 925457822, 10735707149, 125128265025, 1464140655619, 17188834766497, 202366206841241, 2388313959181973, 28246993739096305, 334711010978735163, 3972765235517468758, 47224110710958716845

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A108081, A371742, A371743.

a(n) = sum(k=0, n\2, binomial(5*n-k, n-2*k));

a(n) = [x^n] 1/((1-x-x^2) * (1-x)^(4*n)).

a(n) ~ 5^(5*n + 3/2) / (19 * sqrt(Pi*n) * 2^(8*n - 1/2)). - Vaclav Kotesovec, Apr 05 2024

A133915 a(n) = Sum_{i=0..n} C(2n-i,n+i)2^i.

Original entry on oeis.org

1, 2, 8, 30, 116, 452, 1772, 6974, 27524, 108852, 431168, 1709996, 6788536, 26971856, 107235668, 426594110, 1697855876, 6760326116, 26927208368, 107288242820, 427596003416, 1704598377176, 6796820059928, 27106584400460, 108123625907816, 431355955330952

Offset: 0

Paul Barry, Sep 28 2007

A transform of the Jacobsthal numbers A001045(n+1) under the mapping g(x)->(1/(c(x)*sqrt(1-4*x)))*g(x*c(x)), c(x) the g.f. of A000108. Hankel transform is A001787(n+1).

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000108, A001787, A108081.

CoefficientList[Series[(1-4*x+(1-x)*Sqrt[1-4*x])/((x+2)*(1-4*x)^(3/2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

a(n) = sum(i=0, n, binomial(2*n-i, n+i)*2^i); \\ Michel Marcus, Jul 08 2021

G.f.: (1-4*x+(1-x)*sqrt(1-4*x))/((x+2)*(1-4*x)^(3/2)).
a(n) = Sum_{k=0..n} C(2*n-k,n+k)*2^k.
a(n) = Sum_{k=0..n} C(n+k-1,k)*A001045(n-k+1).
2*n*a(n) +3*(4-5*n)*a(n-1) +6*(4*n-7)*a(n-2) + 8*(2*n-3)*a(n-3)=0. - R. J. Mathar, Nov 14 2011
a(n) ~ 4^n/3. - Vaclav Kotesovec, Oct 20 2012

A360149 a(n) = Sum_{k=0..floor(n/2)} binomial(2n+k,n-2k).

Original entry on oeis.org

1, 2, 7, 27, 107, 429, 1731, 7012, 28478, 115864, 471991, 1924483, 7852083, 32053208, 130893949, 534673600, 2184482707, 8926392419, 36479840422, 149095843951, 609400587426, 2490900041118, 10181669553847, 41618414303969, 170118507902985, 695366323719302

Offset: 0

Seiichi Manyama, Jan 28 2023

nonn

Cf. A014300, A106188, A108081, A114121, A176287.

Cf. A000108, A360144, A360150.

A360149 := proc(n)
    add(binomial(2*n+k,n-2*k),k=0..floor(n/2)) ;
end proc:
seq(A360149(n),n=0..40) ; # R. J. Mathar, Mar 02 2023

a[n_] := Sum[Binomial[2*n + k, n - 2*k], {k, 0, Floor[n/2]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)

a(n) = sum(k=0, n\2, binomial(2*n+k, n-2*k));

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^2*(2/(1+sqrt(1-4*x)))^5)))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^2 * c(x)^5) ), where c(x) is the g.f. of A000108.
a(n) ~ sqrt((7 - 5*(2/(173 + 21*sqrt(69)))^(1/3) + ((173 + 21*sqrt(69))/2)^(1/3)) / 69) / ((4 - (2/(25 - 3*sqrt(69)))^(1/3) - ((25 - 3*sqrt(69))/2)^(1/3))/3)^n. - Vaclav Kotesovec, Jan 28 2023
D-finite with recurrence n*(47*n-1011)*a(n) +(-261*n^2 +8567*n -6378)*a(n-1) +2*(-165*n^2 -9388*n +16143)*a(n-2) +(3089*n^2 +919*n -27492)*a(n-3) +2*(-1283*n^2 +3900*n +3981)*a(n-4) +4*(81*n+11)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Mar 02 2023

cp's OEIS Frontend

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

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A371742 a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-k,n-2*k).

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A159965 Riordan array (1/sqrt(1-4x), (1-2x-(1-3x)c(x))/(x*sqrt(1-4x))), c(x) the g.f. of A000108.

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A371743 a(n) = Sum_{k=0..floor(n/2)} binomial(4*n-k,n-2*k).

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A371744 a(n) = Sum_{k=0..floor(n/2)} binomial(5*n-k,n-2*k).

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A133915 a(n) = Sum_{i=0..n} C(2*n-i,n+i)*2^i.

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Mathematica

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A360149 a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k,n-2*k).

Views

Author

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Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A371742 a(n) = Sum_{k=0..floor(n/2)} binomial(3n-k,n-2k).

A371743 a(n) = Sum_{k=0..floor(n/2)} binomial(4n-k,n-2k).

A371744 a(n) = Sum_{k=0..floor(n/2)} binomial(5n-k,n-2k).

A133915 a(n) = Sum_{i=0..n} C(2n-i,n+i)2^i.

A360149 a(n) = Sum_{k=0..floor(n/2)} binomial(2n+k,n-2k).