A374848 -id:A374848 - cp's OEIS frontend

A082480 a(n) = Product_{k=1..n} (F(k)+1) where F(k) denotes the k-th Fibonacci number.

1, 2, 4, 12, 48, 288, 2592, 36288, 798336, 27941760, 1564738560, 140826470400, 20419838208000, 4778242140672000, 1806175529174016000, 1103573248325323776000, 1090330369345419890688000, 1742347930213980985319424000, 4503969399603140847050711040000

Offset: 0

Benoit Cloitre, Apr 27 2003

nonn

Equals row sums (unsigned) of triangle A158472. - Gary W. Adamson, Mar 20 2009

Cf. A000045, A158472. - Gary W. Adamson, Mar 20 2009

with(combinat): a:= n->mul(fibonacci(j)+1, j=0..n): seq(a(n), n=0..20); # Zerinvary Lajos, Mar 29 2009

Table[Product[Fibonacci[k]+1,{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 19 2015 *)

PARI
```
a(n)=prod(k=1,n,fibonacci(k)+1)
```

a(n) ~ f * C * ((1+sqrt(5))/2)^(n*(n+1)/2) / 5^(n/2), where C = A062073 = 1.2267420107203532444176302304553616558714096904402504196432973... is the Fibonacci factorial constant and f = Product_{k>=1} (1 + 1/Fibonacci(k)) = 13.150966657784184367612433370626658932190199543164284701354100747157698046... . - Vaclav Kotesovec, Jul 19 2015

Equals the obverse convolution of A000012 and A000045; see A374848. a(n) = (F(n)+1)*a(n-1) for n>=1, where F(n) = A000045(n) = n-th Fibonacci number. - Clark Kimberling, Aug 05 2024

A374880 Obverse convolution (floor(3n/2))**(floor(3n/2)); see Comments.

Original entry on oeis.org

0, 1, 18, 256, 5400, 117649, 3359232, 100000000, 3643149312, 137858491849, 6126151500000, 281474976710656, 14777503265582208, 799006685782884121, 48413259982080000000, 3011361496339065143296, 206882551397716479442944, 14551915228366851806640625

Offset: 0

Clark Kimberling, Jul 31 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences. If k>=0, then a(2k) is even and a(2k+1) is a square.

Cf. A016789, A374848.

s[n_] := Floor[3 n/2]; t[n_] := Floor[3 n/2];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[u[n], {n, 0, 24}]

a(n) ~ exp(-1/3) * (3*n/2)^(n+1). - Vaclav Kotesovec, Aug 02 2024

A374881 Obverse convolution (n)**(n^2); see Comments.

Original entry on oeis.org

0, 1, 16, 405, 15360, 818125, 58226688, 5332085577, 610140160000, 85235284359225, 14264819712000000, 2815701027697558429, 646960843646287478784, 171112492588968115453125, 51595090958399913852928000, 17587698619968027952119140625

Offset: 0

Clark Kimberling, Jul 31 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences.

Cf. A000027, A000290, A374848, A376523, A376524.

s[n_] := n; t[n_] := n^2;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[u[n], {n, 0, 18}]

a(n) ~ n^(2*n + 1) / exp(2*n + 1 - Pi*sqrt(n)). - Vaclav Kotesovec, Jul 31 2024

A375041 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = x+1. See Comments.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 8, 17, 10, 1, 18, 97, 180, 100, 1, 35, 403, 1829, 3160, 1700, 1, 61, 1313, 12307, 50714, 83860, 44200, 1, 98, 3570, 60888, 506073, 1960278, 3147020, 1635400, 1, 148, 8470, 239388, 3550473, 27263928, 101160920, 158986400, 81770000, 1, 213

Offset: 1

Clark Kimberling, Sep 11 2024

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

			First 3 polynomials in s(x)**t(x) are
  1 + x,
  1 + 3 x + 2 x^2,
  1 + 8 x + 17 x^2 + 10 x^3.
First 5 rows of array:
  1    1
  1    3     2
  1    8    17    10
  1   18    97   180   100
  1   35  4034  1829  3160  1700

Cf. A000290, A081489 (column 2), A101686 (T(n,n+1)), A374848, A375042, A375043.

s[n_] := n^2  x; t[n_] := 1 + x;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)

A375042 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = 2x+1. See Comments.

Original entry on oeis.org

1, 2, 1, 5, 6, 1, 11, 36, 36, 1, 22, 157, 432, 396, 1, 40, 553, 3258, 8172, 7128, 1, 67, 1633, 18189, 96138, 227772, 192456, 1, 105, 4179, 80243, 787320, 3881016, 8847792, 7313328, 1, 156, 9534, 293372, 4879713, 44034336, 206779608, 458550720, 372979728, 1

Offset: 1

Clark Kimberling, Sep 11 2024

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

			First 3 polynomials in s(x)**t(x) are
  1 + 2x,
  1 + 5 x + 6 x^2,
  1 + 11 x + 36 x^2 + 36 x^3.
First 5 rows of array:
  1   2
  1   5      6
  1   11    36    36
  1   22   157   432   396
  1   40   553  3258  8172  7128

Cf. A000290, A277355 ((1/2)T(n,n+1)), A374848, A375041, A375043.

s[n_] := n^2  x; t[n_] := 1 + 2 x;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]    (* sequence *)

A375043 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = x+2. See Comments.

Original entry on oeis.org

2, 1, 4, 6, 2, 8, 32, 34, 10, 16, 144, 388, 360, 100, 32, 560, 3224, 7316, 6320, 1700, 64, 1952, 21008, 98456, 202856, 167720, 44200, 128, 6272, 114240, 974208, 4048584, 7841112, 6294040, 1635400, 256, 18944, 542080, 7660416, 56807568, 218111424, 404643680

Offset: 1

Clark Kimberling, Sep 14 2024

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

			First 3 polynomials in s(x)**t(x) are
2 + x,
4 + 6 x + 2 x^2,
8 + 32 x + 34 x^2 + 10 x^3.
First 5 rows of array:
 2    1
 4    6     2
 8   32    34    10
16  144   388   360   100
32  560  3224  7316  6320  1700

Cf. A000290, A101686 (T(n,n+1)), A374848, A375041, A375042.

s[n_] := n^2  x; t[n_] := x + 2;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)

A375047 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = x+1 and t(x) = F(n) = n-th Fibonacci number. See Comments.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 8, 5, 1, 12, 28, 23, 8, 1, 48, 124, 120, 55, 12, 1, 288, 792, 844, 450, 127, 18, 1, 2592, 7416, 8388, 4894, 1593, 289, 27, 1, 36288, 106416, 124848, 76904, 27196, 5639, 667, 41, 1, 798336, 2377440, 2853072, 1816736, 675216, 151254, 20313

Offset: 1

Clark Kimberling, Sep 15 2024

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

			First 3 polynomials in s(x)**t(x) are
1 + x,
2 + 3 x + x^2,
4 + 8 x + 5 x^2 + x^3.
First 5 rows of array:
 1    1
 2    3    1
 4    8    5   1
12   28   23   8   1
48  124  120  55  12  1

Cf. A000045, A232896 (T(n,n)), A374848, A375048, A375049.

s[n_] := x + 1; t[n_] := Fibonacci[n];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)

A375048 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = x+F(n) and t(x) = F(n), where F(n) = n-th Fibonacci number (A000045). See Comments.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 5, 4, 1, 16, 32, 24, 8, 1, 162, 297, 216, 78, 14, 1, 3600, 5640, 3649, 1248, 238, 24, 1, 147456, 196608, 110848, 34240, 6256, 676, 40, 1, 12320100, 13667940, 6521589, 1746426, 286843, 29568, 1867, 66, 1, 2058386904, 1878686460, 746158770

Offset: 1

Clark Kimberling, Sep 15 2024

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

			First 3 polynomials in s(x)**t(x) are
0 + x,
1 + 2 x + x^2,
2 + 4 x + 4 x^2 + x^3.
First 5 rows of array:
  0    1
  1    2    1
  2    5    4   1
 16   32   24   8   1
162  297  216  78  14  1

Cf. A000045, A019274 (T(n,n)), A374848, A375047, A375049.

s[n_] := x + Fibonacci[n]; t[n_] := Fibonacci[n];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)

A375049 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = x+F(n) and t(x) = x+F(n), and F(n) = n-th Fibonacci number (A000045). See Comments.

Original entry on oeis.org

0, 2, 1, 4, 4, 2, 10, 16, 8, 16, 64, 96, 64, 16, 162, 594, 864, 624, 224, 32, 3600, 11280, 14596, 9984, 3808, 768, 64, 147456, 393216, 443392, 273920, 100096, 21632, 2560, 128, 12320100, 27335880, 26086356, 13971408, 4589488, 946176, 119488, 8448, 256

Offset: 1

Clark Kimberling, Jul 31 2024

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays. If n is odd, then the polynomial u(n) is a square. Every T(n,k) except T(2,1) is even.

			First 3 polynomials in s(x)**t(x) are
  0 + 2x,
  1 + 4 x + 4x^2,
  2 + 10 x + 16 x^2 + 8 x^3.
First 5 rows of array:
  0   2
  1   4   4
  2   10  16   8
  16  64  96  64  16
  162 594 864 624 224 32

Cf. A000045, A000079 (T(n,n+1)), A374848, A375047, A375048.

s[n_] := x + Fibonacci[n]; t[n_] := Fibonacci[n];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]    (* sequence *)

A374877 Obverse convolution (3n+1)**(3n+1); see Comments.

Original entry on oeis.org

2, 25, 512, 14641, 537824, 24137569, 1280000000, 78310985281, 5429503678976, 420707233300201, 36028797018963968, 3379220508056640625, 344498040522809827328, 37929227194915558802161, 4485286068729022118887424, 566977372488557307219621121

Offset: 0

Clark Kimberling, Sep 13 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences.

If k>=0, then a(2k) is even and a(2k+1) is a square.

Cf. A016777, A374848, A374878.

s[n_] := 3 n + 1; t[n_] := 3 n + 1;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[u[n], {n, 0, 17}]
(* or *)
Table[(3*n+2)^(n+1), {n,0,20}] (* Vaclav Kotesovec, Sep 13 2024 *)

From Vaclav Kotesovec, Sep 13 2024: (Start)
a(n) = (3*n+2)^(n+1).
a(n) ~ exp(2/3) * 3^(n+1) * n^(n+1). (End)

cp's OEIS Frontend

A082480 a(n) = Product_{k=1..n} (F(k)+1) where F(k) denotes the k-th Fibonacci number.

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A374880 Obverse convolution (floor(3n/2))**(floor(3n/2)); see Comments.

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A374881 Obverse convolution (n)**(n^2); see Comments.

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A375041 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = x+1. See Comments.

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A375042 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = 2x+1. See Comments.

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A375043 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = x+2. See Comments.

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A375047 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = x+1 and t(x) = F(n) = n-th Fibonacci number. See Comments.

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A375048 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = x+F(n) and t(x) = F(n), where F(n) = n-th Fibonacci number (A000045). See Comments.

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A375049 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = x+F(n) and t(x) = x+F(n), and F(n) = n-th Fibonacci number (A000045). See Comments.

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