A045925 -id:A045925 - cp's OEIS frontend

A086926 Product of Fibonacci and (shifted) triangular numbers.

0, 0, 1, 6, 18, 50, 120, 273, 588, 1224, 2475, 4895, 9504, 18174, 34307, 64050, 118440, 217192, 395352, 714951, 1285350, 2298660, 4091241, 7250221, 12797568, 22507500, 39452725, 68942718, 120132558, 208776974, 361937400, 626015085, 1080441264

Offset: 0

James FitzSimons (cherry(AT)getnet.net), Sep 20 2003

Colin Barker, Table of n, a(n) for n = 0..1000
Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, The Fibonacci-Fubini and Lucas-Fubini numbers, arXiv:2407.04409 [math.CO], 2024. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. A000045, A000217, A045925.

Array[Fibonacci[#] PolygonalNumber[# - 1] &, 33, 0] (* or *)
LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 1, 6, 18, 50}, 33] (* or *)
CoefficientList[Series[x^2*(1 + 3 x + x^3)/(1 - x - x^2)^3, {x, 0, 32}], x] (* Michael De Vlieger, Dec 17 2017 *)

numlib::fibonacci(n)*binomial(n,2) $ n = 0..35; // Zerinvary Lajos, May 09 2008

concat(vector(2), Vec(x^2*(1 + 3*x + x^3) / (1 - x - x^2)^3 + O(x^40))) \\ Colin Barker, Sep 20 2017

From Franklin T. Adams-Watters, Feb 03 2006: (Start)
a(n) = A000045(n)*A000217(n-1) = A000045(n)*n*(n-1)/2.
a(n) = (n/(n-2)*a(n-1) + n*(n-1))/((n-2)*(n-3))*a(n-2).
G.f.: x^2*(1+3x+x^3)/(1-x-x^2)^3. (End)
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} i * C(n-k-1,k). - Wesley Ivan Hurt, Sep 19 2017
From Colin Barker, Sep 20 2017: (Start)
a(n) = ((-1)*(2^(-1-n)*((1-sqrt(5))^n - (1+sqrt(5))^n)*(-1+n)*n)) / sqrt(5).
a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6) for n>5. (End)
a(n) = A001629(n-2)+3*A001629(n-3)+A001629(n-5). - R. J. Mathar, May 16 2025

Definition and more terms from Franklin T. Adams-Watters, Feb 03 2006

A259546 a(n) = n^3*Fibonacci(n).

Original entry on oeis.org

0, 1, 8, 54, 192, 625, 1728, 4459, 10752, 24786, 55000, 118459, 248832, 511901, 1034488, 2058750, 4042752, 7846061, 15069888, 28677479, 54120000, 101370906, 188586728, 348669719, 640991232, 1172265625, 2133603368, 3866095494, 6976587072, 12541531081

Offset: 0

Colin Barker, Jun 30 2015

Colin Barker, Table of n, a(n) for n = 0..1000
Prabha Sivaraman Nair and Rejikumar Karunakaran, On k-Fibonacci Brousseau Sums, J. Int. Seq. (2024) Art. No. 24.6.4. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,5,8,-2,-4,-1).

Cf. A000045, A000578, A045925, A259451, A259547.

a:= n-> n^3*(<<1|1>, <1|0>>^n)[1, 2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 30 2015

Array[#^3*Fibonacci[#] &, 50, 0] (* Paolo Xausa, Jul 15 2024 *)

PARI
```
a(n) = n^3*fibonacci(n)
```

concat(0, Vec(x*(x^2+1)*(x^4-4*x^3+23*x^2+4*x+1)/(x^2+x-1)^4 + O(x^50)))

G.f.: x*(x^2+1)*(x^4-4*x^3+23*x^2+4*x+1) / (x^2+x-1)^4.
Sum_{k=1..n} a(k) = (n^3-6*n^2+24*n-50)*A000045(n+1) + ((n+1)^3-6*(n+1)^2+24*(n+1)-50)*A000045(n) + 50. - Prabha Sivaramannair, Jul 15 2024
E.g.f.: exp(x/2)*x*(5*(1 + x)*(1 + 2*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(1 + x*(9 + 4*x))*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Aug 25 2024

A259547 a(n) = n^4*Fibonacci(n).

Original entry on oeis.org

0, 1, 16, 162, 768, 3125, 10368, 31213, 86016, 223074, 550000, 1303049, 2985984, 6654713, 14482832, 30881250, 64684032, 133383037, 271257984, 544872101, 1082400000, 2128789026, 4148908016, 8019403537, 15383789568, 29306640625, 55473687568, 104384578338

Offset: 0

Colin Barker, Jun 30 2015

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-5,-10,15,11,-15,-10,5,5,1).

Cf. A000045, A000583, A045925, A259451, A259546.

a:= n-> n^4*(<<1|1>, <1|0>>^n)[1, 2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 30 2015

Table[n^4 Fibonacci[n],{n,0,30}] (* or *) LinearRecurrence[{5,-5,-10,15,11,-15,-10,5,5,1},{0,1,16,162,768,3125,10368,31213,86016,223074},30] (* Harvey P. Dale, Mar 09 2016 *)

PARI
```
a(n) = n^4*fibonacci(n)
```

concat(0, Vec(-x*(x^8 -11*x^7 +87*x^6 -48*x^5 +240*x^4 +48*x^3 +87*x^2 +11*x +1)/(x^2 +x -1)^5 + O(x^50)))

G.f.: -x*(x^8-11*x^7+87*x^6-48*x^5+240*x^4+48*x^3+87*x^2+11*x+1) / (x^2+x-1)^5.

E.g.f.: exp(x/2)*x*(5*(1 + 7*x + 12*x^2 + 3*x^3)*cosh(sqrt(5)*x/2) + sqrt(5)*(1 + 21*x + 24*x^2 + 7*x^3)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Aug 25 2024

A317403 a(n)=(-1)^((n-2)(n-1)/2)2^(n-1)*n^(n-3).

Original entry on oeis.org

1, 1, -4, -32, 400, 6912, -153664, -4194304, 136048896, 5120000000, -219503494144, -10567230160896, 564668382613504, 33174037869887488, -2125764000000000000, -147573952589676412928, 11034809241396899282944, 884295678882933431599104, -75613185918270483380568064

Offset: 1

Rigoberto Florez, Aug 26 2018

sign

Discriminant of Fibonacci polynomials.

Fibonacci polynomials are defined as F(0)=0, F(1)=1 and F(n)=x*F(n-1)+F(n-2) for n>1. Coefficients are given in triangle A168561 with offset 1.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, 2018.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A168561, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

Essentially the same as A127670.

[(-1)^((n-2)*(n-1) div 2)*2^(n-1)*n^(n-3): n in [1..20]]; // Vincenzo Librandi, Aug 27 2018

Array[(-1)^((#-2)*(#-1)/2)*2^(#-1)*#^(#-3)&,20]

concat([1], [poldisc(p) | p<-Vec(x/(1-x^2-y*x) - x + O(x^20))]) \\ Andrew Howroyd, Aug 26 2018

A104731 Triangle T(n,k) = sum_{j=k..n} (j+1)*binomial(k,j-k), read by rows, 0<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 5, 11, 4, 1, 5, 16, 19, 5, 1, 5, 16, 37, 29, 6, 1, 5, 16, 44, 71, 41, 7, 1, 5, 16, 44, 103, 121, 55, 8, 1, 5, 16, 44, 112, 211, 190, 71, 9, 1, 5, 16, 44, 112, 261, 390, 281, 89, 10, 1, 5, 16, 44, 112, 272, 555, 666, 397, 109, 11

Offset: 0

Gary W. Adamson, Mar 20 2005

			The first few rows of the triangle are:
1;
1, 2;
1, 5, 3;
1, 5, 11, 4
1, 5, 16, 19, 5;
1, 5, 16, 37, 29, 6;
...

Cf. A014286 (row sums), A045925, A026729.

Product of the triangles A(n,k) = k+1 and B = binomial(k,n-k) = [1; 0, 1; 0, 1, 1; 0, 0, 2, 1; 0, 0, 1, 3, 1;...], the triangular view of A026729.

A113684 Expansion of x(1-x^2-x^3)/((1-x)(1-x-x^2))^2.

Original entry on oeis.org

0, 1, 4, 11, 25, 52, 102, 193, 356, 645, 1153, 2040, 3580, 6241, 10820, 18671, 32089, 54956, 93826, 159745, 271300, 459721, 777409, 1312176, 2211000, 3719617, 6248452, 10482323, 17562841, 29391460, 49132638, 82048705, 136884260

Offset: 0

Paul Barry, Nov 05 2005

Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,4,0,-1)

Cf. A045925, A006478, A023610.

a(n)=4a(n-1)-4a(n-2)-2a(n-3)+4a(n-4)-a(n-6); a(n)=sum{k=0..n, (n-k)*C(n-k, k+1)}; a(n)=n*(F(n+2)-1)-(1+((n-5)*F(n-1)+(3n-8)*F(n))/5).

A137392 (10-n) * Fibonacci(n).

Original entry on oeis.org

9, 8, 14, 18, 25, 32, 39, 42, 34, 0, -89, -288, -699, -1508, -3050, -5922, -11179, -20672, -37629, -67650, -120406, -212532, -372541, -649152, -1125375, -1942288, -3339106, -5720598, -9770351, -16640800, -28271649

Offset: 1

Roger L. Bagula and Gary W. Adamson, Apr 10 2008

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045, A045925.

Table[(10-n)Fibonacci[n],{n,40}] (* or *) LinearRecurrence[{2,1,-2,-1},{9,8,14,18},40] (* Harvey P. Dale, Sep 05 2022 *)

PARI
```
a(n)=(10-n)*fibonacci(n)
```

G.f.: x*(-11*x^2 - 10*x + 9) / (x^2+x-1)^2. - Colin Barker, Jan 01 2013

Edited by Ralf Stephan, Dec 24 2013

A317450 a(n)=(-1)^((n-2)(n-1)/2)2^((n-1)^2)*n^(n-3).

Original entry on oeis.org

1, 1, -16, -2048, 1638400, 7247757312, -164995463643136, -18446744073709551616, 9803356117276277820358656, 24178516392292583494123520000000, -271732164163901599116133024293512544256, -13717048991958695477963985711266803110069141504, 3074347100178259797134292590832254504315406543889629184

Offset: 1

Rigoberto Florez, Aug 26 2018

sign

Discriminant of Pell polynomials.

Pell polynomials are defined as P(0)=0, P(1)=1 and P(n)=2xP(n-1)+P(n-2) for n>1.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, 2018.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Pell Polynomial

Cf. A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

Essentially the same as A086804.

Array[(-1)^((#-2)*(#-1)/2)* 2^((#-1)^2)*#^(#-3)&,15]

cp's OEIS Frontend

A086926 Product of Fibonacci and (shifted) triangular numbers.

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A259546 a(n) = n^3*Fibonacci(n).

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A259547 a(n) = n^4*Fibonacci(n).

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A317403 a(n)=(-1)^((n-2)*(n-1)/2)*2^(n-1)*n^(n-3).

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A104731 Triangle T(n,k) = sum_{j=k..n} (j+1)*binomial(k,j-k), read by rows, 0<=k<=n.

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A113684 Expansion of x(1-x^2-x^3)/((1-x)(1-x-x^2))^2.

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A137392 (10-n) * Fibonacci(n).

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A317450 a(n)=(-1)^((n-2)*(n-1)/2)*2^((n-1)^2)*n^(n-3).

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A317403 a(n)=(-1)^((n-2)(n-1)/2)2^(n-1)*n^(n-3).

A317450 a(n)=(-1)^((n-2)(n-1)/2)2^((n-1)^2)*n^(n-3).