A141611 -id:A141611 - cp's OEIS frontend

A109906 A triangle based on A000045 and Pascal's triangle: T(n,m) = Fibonacci(n-m+1) * Fibonacci(m+1) * binomial(n,m).

1, 1, 1, 2, 2, 2, 3, 6, 6, 3, 5, 12, 24, 12, 5, 8, 25, 60, 60, 25, 8, 13, 48, 150, 180, 150, 48, 13, 21, 91, 336, 525, 525, 336, 91, 21, 34, 168, 728, 1344, 1750, 1344, 728, 168, 34, 55, 306, 1512, 3276, 5040, 5040, 3276, 1512, 306, 55, 89, 550, 3060, 7560, 13650, 16128, 13650, 7560, 3060, 550, 89

Offset: 0

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Row sums give A081057.

			Triangle T(n,k) begins:
   1;
   1,   1;
   2,   2,    2;
   3,   6,    6,    3;
   5,  12,   24,   12,     5;
   8,  25,   60,   60,    25,     8;
  13,  48,  150,  180,   150,    48,    13;
  21,  91,  336,  525,   525,   336,    91,   21;
  34, 168,  728, 1344,  1750,  1344,   728,  168,   34;
  55, 306, 1512, 3276,  5040,  5040,  3276, 1512,  306,  55;
  89, 550, 3060, 7560, 13650, 16128, 13650, 7560, 3060, 550, 89;
  ...

Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Peter McCalla, Asamoah Nkwanta, Catalan and Motzkin Integral Representations, arXiv:1901.07092 [math.NT], 2019.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A141611, A141617, A000045, A081057.

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A111006, A114197, A162741, A228074.

a109906 n k = a109906_tabl !! n !! k
a109906_row n = a109906_tabl !! n
a109906_tabl = zipWith (zipWith (*)) a058071_tabl a007318_tabl
-- Reinhard Zumkeller, Aug 15 2013

f:= n-> combinat[fibonacci](n+1):
T:= (n, k)-> binomial(n, k)*f(k)*f(n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 26 2023

Clear[t, n, m] t[n_, m_] := Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]*Binomial[n, m]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

T(n,m) = Fibonacci(n-m+1)*Fibonacci(m+1)*binomial(n,m).

T(n,k) = A058071(n,k) * A007318(n,k). - Reinhard Zumkeller, Aug 15 2013

Offset changed by Reinhard Zumkeller, Aug 15 2013

A007466 Exponential-convolution of natural numbers with themselves.

Original entry on oeis.org

1, 4, 14, 44, 128, 352, 928, 2368, 5888, 14336, 34304, 80896, 188416, 434176, 991232, 2244608, 5046272, 11272192, 25034752, 55312384, 121634816, 266338304, 580911104, 1262485504, 2734686208, 5905580032, 12717129728

Offset: 1

N. J. A. Sloane

Define a triangle T by T(n,1) = n*(n-1)+1 and T(r,c) = T(r,c-1) + T(r-1,c-1), then a(n) = T(n,n). - J. M. Bergot, Mar 03 2013
From David Callan, Jul 11 2014: (Start)
With offset 0, a(n) is the number of 2 X n 0-1 matrices that do not contain
  1 1    0 0
  0 0 or 1 1, as a 2 X 2 submatrix,
See Ju and Seo link, Theorem 3.2. (End)
a(n) is the sum of all ways of adding the k-tuples of the terms in the (n-1)-st row of Pascal's triangle A007318. For n=4 take row 3 of A007318: 1,3,3,1, giving (1)+(3)+(3)+(1)=8; (1+3)+(3+3)+(3+1)=14; (1+3+3)+(3+3+1)=14; (1+3+3+1)=8. The sum of these four terms is 8+14+14+8=44. - J. M. Bergot, Jun 17 2017
Binomial transform of A002061. - Jules Beauchamp, Jan 04 2022
a(n+1) is the number of strings of length n defined on {0,1,2,3} that contain at most one 2, at most one 3, and have no restriction on the number of 0s and 1s. For example, for n=2, a(3)=14 since from the 16 strings of length 2 we exclude 22 and 33. - Enrique Navarrete, May 03 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of 0/1-matrices avoiding some 2x2 matrices, arXiv:1107.1299 [math.CO], 2011.
Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices, Discrete Math., 312 (2012), 2473-2481.
N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A002061, A007318, A141611, A228643.

a007466 n = a228643 n n  -- Reinhard Zumkeller, Aug 29 2013

[2^(n-1)*(n+(n-1)*(n-2)/4) : n in [1..30]]; // Wesley Ivan Hurt, Jul 11 2014

A007466:=n->2^(n-1)*n+1/4*2^(n-1)*(n-1)*(n-2): seq(A007466(n), n=1..30);

Table[2^(n-1)*(n + (n-1)*(n-2)/4), {n, 30}] (* Wesley Ivan Hurt, Jul 11 2014 *)

def A007466(n): return 2^(n-3)*(n^2+n+2)
[A007466(n) for n in range(1,31)] # G. C. Greubel, Sep 22 2024

E.g.f.: (Sum_{n >= 1} n*x^(n-1)/(n-1)!)^2.
a(n) = 2^(n-1)*n + 2^(n-3)*(n-1)*(n-2).
a(n) = Sum_{k=0..(n+2)} C(n+2, k) * floor(k/2)^2. - Paul Barry, Mar 06 2003
E.g.f.: (1+x)^2*exp(2*x). - Vladeta Jovovic, Sep 09 2003
G.f.: x*(1 - 2*x + 2*x^2)/(1-2*x)^3. - Vladimir Kruchinin, Sep 28 2011
E.g.f.: U(0) where U(k)= 1 + 2*x/( 1 - x/(2 + x - 4/( 2 + x*(k+1)/U(k+1)))) ; (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Oct 28 2012
a(n) = A228643(n, n). - Reinhard Zumkeller, Aug 29 2013
a(n) = Sum_{k=0..n-1} A141611(n-1, k). - G. C. Greubel, Sep 22 2024

A110023 A triangle of coefficients based on A000931 and Pascal's triangle: a(n)=a(n-2)+a(n-3); t(n,m)=a(n - m + 1)a(m + 1)Binomial[n, m].

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 6, 6, 2, 3, 8, 24, 8, 3, 4, 15, 40, 40, 15, 4, 5, 24, 90, 80, 90, 24, 5, 7, 35, 168, 210, 210, 168, 35, 7, 9, 56, 280, 448, 630, 448, 280, 56, 9, 12, 81, 504, 840, 1512, 1512, 840, 504, 81, 12, 16, 120, 810, 1680, 3150, 4032, 3150, 1680, 810, 120, 16

Offset: 1

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Row sums are:

{1, 2, 6, 16, 46, 118, 318, 840, 2216, 5898, 15584}

			{1},
{1, 1},
{2, 2, 2},
{2, 6, 6, 2},
{3, 8, 24, 8, 3},
{4, 15, 40, 40, 15, 4},
{5, 24, 90, 80, 90, 24, 5},
{7, 35, 168, 210, 210, 168, 35, 7},
{9, 56, 280, 448, 630, 448, 280, 56, 9},
{12, 81, 504, 840, 1512, 1512, 840, 504, 81, 12},
{16, 120, 810, 1680, 3150, 4032, 3150, 1680, 810, 120, 16}

Cf. A141611, A141617, A000931.

Clear[t, a, n, m] a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 2] + a[n - 3]; t[n_, m_] := a[(n - m + 1)]*a[(m + 1)]*Binomial[n, m]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}] Flatten[%]

a(n)=a(n-2)+a(n-3); t(n,m)=a(n - m + 1)*a(m + 1)*Binomial[n, m].

A110102 A triangle of coefficients based on A000931: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 2, 3, 4, 3, 4, 4, 3, 4, 5, 4, 6, 4, 6, 4, 5, 7, 5, 8, 6, 6, 8, 5, 7, 9, 7, 10, 8, 9, 8, 10, 7, 9, 12, 9, 14, 10, 12, 12, 10, 14, 9, 12, 16, 12, 18, 14, 15, 16, 15, 14, 18, 12, 16

Offset: 1

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Row sums are:

{1, 2, 5, 8, 14, 22, 34, 52, 77, 114, 166}

			{1},
{1, 1},
{2, 1, 2},
{2, 2, 2, 2},
{3, 2, 4, 2, 3},
{4, 3, 4, 4, 3, 4},
{5, 4, 6, 4, 6, 4, 5},
{7, 5, 8, 6, 6, 8, 5, 7},
{9, 7, 10, 8, 9, 8, 10, 7, 9},
{12, 9, 14, 10, 12, 12, 10, 14, 9, 12},
{16, 12, 18, 14, 15, 16, 15, 14, 18, 12, 16}

Cf. A141611, A141617, A000931.

Clear[t, a, n, m] a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 2] + a[n - 3]; t[n_, m_] := a[(n - m + 1)]*a[(m + 1)]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1).

A110361 A triangle of coefficients based on A000931 and A000045: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)a(m + 1)Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 6, 4, 4, 6, 15, 6, 16, 6, 15, 32, 15, 24, 24, 15, 32, 65, 32, 60, 36, 60, 32, 65, 147, 65, 128, 90, 90, 128, 65, 147, 306, 147, 260, 192, 225, 192, 260, 147, 306, 660, 306, 588, 390, 480, 480, 390, 588, 306, 660, 1424, 660, 1224, 882, 975, 1024, 975, 882

Offset: 1

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Row sums are:

{1, 2, 9, 20, 58, 142, 350, 860, 2035, 4848, 11354}.

			{1},
{1, 1},
{4, 1, 4},
{6, 4, 4, 6},
{15, 6, 16, 6, 15},
{32, 15, 24, 24, 15, 32},
{65, 32, 60, 36, 60, 32, 65},
{147, 65, 128, 90, 90, 128, 65, 147},
{306, 147, 260, 192, 225, 192, 260, 147, 306},
{660, 306, 588, 390, 480, 480, 390, 588, 306, 660},
{1424, 660, 1224, 882, 975, 1024, 975, 882, 1224, 660, 1424}

Cf. A141611, A141617, A000931, A000045, A058071.

Clear[t, a, n, m] a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 2] + a[n - 3]; t[n_, m_] := a[(n - m + 1)]*a[(m + 1)]*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1)*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].

cp's OEIS Frontend

A109906 A triangle based on A000045 and Pascal's triangle: T(n,m) = Fibonacci(n-m+1) * Fibonacci(m+1) * binomial(n,m).

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A007466 Exponential-convolution of natural numbers with themselves.

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A110023 A triangle of coefficients based on A000931 and Pascal's triangle: a(n)=a(n-2)+a(n-3); t(n,m)=a(n - m + 1)*a(m + 1)*Binomial[n, m].

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A110102 A triangle of coefficients based on A000931: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1).

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A110361 A triangle of coefficients based on A000931 and A000045: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1)*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].

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A110023 A triangle of coefficients based on A000931 and Pascal's triangle: a(n)=a(n-2)+a(n-3); t(n,m)=a(n - m + 1)a(m + 1)Binomial[n, m].

A110361 A triangle of coefficients based on A000931 and A000045: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)a(m + 1)Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].