A037027 -id:A037027 - cp's OEIS frontend

1, 3, 1, 9, 4, 1, 26, 14, 5, 1, 73, 44, 20, 6, 1, 201, 131, 69, 27, 7, 1, 545, 376, 220, 102, 35, 8, 1, 1460, 1052, 665, 349, 144, 44, 9, 1, 3873, 2888, 1937, 1116, 528, 196, 54, 10, 1, 10191, 7813, 5490, 3402, 1788, 768, 259, 65, 11, 1, 26633, 20892, 15240, 10008

Offset: 0

Wolfdieter Lang, Apr 27 2000 and May 08 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is ((Pell(z))^2)/(Fib(z)*(1-x*z*Fib(z))) with Pell(x)=1/(1-2*x-x^2) = g.f. for A000129(n+1) (Pell numbers without 0) and Fib(x)=1/(1-x-x^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).

This is the second member of the family of Riordan-type matrices obtained from the Fibonacci convolution matrix A037027 by repeated application of the partial row sums procedure.

			{1}; {3,1}; {9,4,1}; {26,14,5,1};...
Fourth row polynomial (n=3): p(3,x)= 26+14*x+5*x^2+x^3

Cf. A037027, A000045, A000129. Row sums: A054449.

a(n, m)=sum(A054446(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m)= sum(a(j-1, m)*A037027(n-j, 0), j=m..n) + A054446(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: (((Pell(x))^2)/Fib(x))*(x*Fib(x))^m, m >= 0, with Fib(x) = g.f. A000045(n+1) and Pell(x) = g.f. A000129(n+1).

A057281 Coefficient triangle of polynomials (falling powers) related to Fibonacci convolutions. Companion triangle to A057282.

Original entry on oeis.org

1, 5, 16, 20, 160, 300, 75, 1075, 4850, 6840, 275, 6100, 48175, 159650, 186120, 1000, 31550, 379700, 2168650, 5846700, 5916240, 3625, 153875, 2605175, 22426825, 103057800, 238437900, 215717040, 13125, 720375, 16273875, 195469125

Offset: 0

Wolfdieter Lang, Sep 13 2000

The row polynomials are p(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,..
The k-th convolution of F0(n) := A000045(n+1), n >= 0, (Fibonacci numbers starting with F0(0)=1) with itself is Fk(n) := A037027(n+k,k) = (p(k-1,n)*(n+1)*F0(n+1) + q(k-1,n)*(n+2)*F0(n))/(k!*5^k), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^(k-m),m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A057282(k,m).
a(k,0)= A030191(k), k >= 0.

			k=2: F2(n)=((5*n^2+21*n+16)*F(n+2)+(5*n^2+27*n+34)*F(n+1))/50, F(n)=A000045(n); see A001628.

Cf. A000045, A037027, A057995, A057280, A057282.

A091186 Triangle read by rows, in which n-th row gives expansion of x^n/((1-x)(1-x-x^2)^n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 8, 4, 1, 1, 12, 18, 13, 5, 1, 1, 20, 38, 35, 19, 6, 1, 1, 33, 76, 86, 59, 26, 7, 1, 1, 54, 147, 197, 164, 91, 34, 8, 1, 1, 88, 277, 430, 420, 281, 132, 43, 9, 1, 1, 143, 512, 904, 1014, 792, 447, 183, 53, 10, 1, 1, 232, 932, 1846, 2338, 2087, 1371

Offset: 0

Paul Barry, Dec 25 2003

Riordan array (1/(1-x),x/(1-x-x^2)). - Paul Barry, Sep 13 2006

			Rows begin {1},{1,1},{1,2,1},{1,4,3,1}...

Row sums are A024537. Diagonal sums are A005578. Second column is A000071. Third column is A006478.

Essentially the vertical partial sums of triangle A037027.

G.f.: (1-y-y^2) / [(1-y(1+y+z))(1-y)].
Number triangle T(n,k)=sum{j=0..n-k, sum{i=0..n-k-j, C(k+j-1,j)C(j,n-k-i-j)}}; - Paul Barry, Sep 13 2006
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) - T(n-3,k), T(0,0) = T(1,0) = T(1,1) = T(2,0) = T(2,2) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 20 2014

A160905 Right hand side of Pascal rhombus A059317.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 9, 8, 3, 1, 29, 22, 13, 4, 1, 82, 72, 42, 19, 5, 1, 255, 218, 146, 70, 26, 6, 1, 773, 691, 476, 261, 107, 34, 7, 1, 2410, 2158, 1574, 914, 428, 154, 43, 8, 1, 7499, 6833, 5122, 3177, 1603, 659, 212, 53, 9, 1, 23575, 21612, 16706, 10816, 5867, 2628, 967

Offset: 0

Paul Barry, May 29 2009

Riordan array (1/sqrt((1+x-x^2)*(1-3*x-x^2)), (1-x-x^2-sqrt((1+x-x^2)*(1-3*x-x^2)))/(2*x)). Can be factored as
(1/(1-x-x^2), x/(1-x-x^2))*(1/sqrt(1-4x^2),xc(x^2)) = (1/(1-x^2),x/(1-x^2))*(1/(1-x),x/(1-x))*(1/sqrt(1-4x^2),xc(x^2))
and (1/(1-x^2),x/(1-x^2))*(1/sqrt(1-2x-3x^2),(1-x-sqrt(1-2x-3x^2))/(2x)).
Here, c(x) is the g.f. of the Catalan numbers A000108.
A160905 = A037027*A108044 = abs(A049310)*A007318*A108044 = abs(A049310)*A094531.

			Triangle begins:
    1;
    1,   1;
    4,   2,   1;
    9,   8,   3,  1;
   29,  22,  13,  4,  1;
   82,  72,  42, 19,  5, 1;
  255, 218, 146, 70, 26, 6, 1;
  ...

Left column gives A059345.

Cf. A059317.

Number triangle T(n,k) = Sum_{i=0..n} (Sum_{j=0..n} C((n+j)/2,j)*C(j,i)*(1+(-1)^(n-j))/2)*C(i,(i-k)/2)*(1+(-1)^(i-k))/2;

T(n,k) = Sum_{j=0..n} C((n+j)/2,j)*((1+(-1)^(n-j))/2)*Sum_{i=0..j} C(j,i)*C(i,j-k-i).

A202193 Triangle read by rows: T(n,m) = coefficient of x^n in expansion of (x/(1 - x - x^2 - x^3 - x^4))^m.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 8, 12, 9, 4, 1, 15, 28, 25, 14, 5, 1, 29, 62, 66, 44, 20, 6, 1, 56, 136, 165, 129, 70, 27, 7, 1, 108, 294, 401, 356, 225, 104, 35, 8, 1, 208, 628, 951, 944, 676, 363, 147, 44, 9, 1, 401, 1328, 2211, 2424, 1935, 1176, 553, 200, 54, 10, 1

Offset: 1

Vladimir Kruchinin, Dec 14 2011

From Philippe Deléham, Feb 16 2014: (Start)
As a Riordan array, this is (1/(1 - x - x^2 - x^3 - x^4), x/(1 - x - x^2 - x^3 - x^4)).
T(n,0) = A000078(n+3); T(n+1,1) = A118898(n+4).
Row sums are A103142(n).
Diagonal sums are A077926(n)*(-1)^n.
Tetranacci convolution triangle. (End)

			Triangle begins:
   1;
   1,  1;
   2,  2,  1;
   4,  5,  3,  1;
   8, 12,  9,  4,  1;
  15, 28, 25, 14,  5,  1;
  29, 62, 66, 44, 20,  6,  1;

Similar sequences : A037027 (Fibonacci convolution triangle), A104580 (tribonacci convolution triangle). - Philippe Deléham, Feb 16 2014

T(n,m):=if n=m then 1 else sum(sum((-1)^i*binomial(k,k-i)*binomial(n-m-4*i-1,k-1),i,0,(n-m-k)/4)*binomial(k+m-1,m-1),k,1,n-m);

T(n,m) = Sum_{k=1..n-m} (Sum_{i=0..floor((n-m-k)/4)} (-1)^i*binomial(k,k-i)*binomial(n-m-4*i-1,k-1))*binomial(k+m-1,m-1), n > m, T(n,n)=1.
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-3,k) + T(n-4,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Feb 16 2014
G.f. for column m: (x/(1 - x - x^2 - x^3 - x^4))^m. - Jason Yuen, Feb 17 2025

A261507 Fibonacci-numbered rows of Pascal's triangle. Triangle read by rows: T(n,k)= binomial(Fibonacci(n), k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 10, 10, 5, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1, 1, 21, 210, 1330, 5985, 20349, 54264, 116280, 203490, 293930, 352716, 352716, 293930, 203490, 116280, 54264, 20349, 5985, 1330, 210, 21, 1

Offset: 0

Maghraoui Abdelkader, Aug 22 2015

Subsequence of A007318.

			1,
1,  1,
1,  1,
1,  2,  1,
1,  3,  3,   1,
1,  5, 10,  10,   5,    1,
1,  8, 28,  56,  70,   56,   28,    8,    1,
1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1

Jean-François Alcover, Table of n, a(n) for n = 0..388 (corrected by _N. J. A. Sloane_, Jan 18 2019)

Cf. A000045, A007318.

Cf. A036355, A037027, A228074, A162741, A034871, A034870, A191797.

Table[Binomial[Fibonacci[n], k], {n, 0, 8}, {k, 0, Fibonacci[n]}]//Flatten (* Jean-François Alcover, Nov 12 2015*)

v = vector(101,j,fibonacci(j)); i=0; n=0; while(n<100, for(k=0, n, print1(binomial(n, k), ", ","")); print(); i=i+1; n=v[i] ;)

T(n, k) = binomial(fibonacci(n), k).
T(n, 1) = fibonacci(n) = A000045(n).
T(n, 2) = A191797(n) for n>3.

A123262 Fibonacci-tribonacci triangle.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 3, 0, 0, 0, 0, 2, 5, 0, 0, 0, 0, 0, 5, 8, 0, 0, 0, 0, 0, 1, 10, 13, 0, 0, 0, 0, 0, 0, 3, 20, 21, 0, 0, 0, 0, 0, 0, 0, 9, 38, 34, 0, 0, 0, 0, 0, 0, 0, 1, 22, 71, 55, 0, 0, 0, 0, 0, 0, 0, 0, 4, 51, 130, 89, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 111, 235, 144

Offset: 0

Philippe Deléham, Nov 06 2006

Or, skew Jacobsthal-Lucas triangle, read by rows.

			Triangle begins:
.0;
.0, 1;
.0, 0, 1;
.0, 0, 0, 2;
.0, 0, 0, 1, 3;
.0, 0, 0, 0, 2, 5;
.0, 0, 0, 0, 0, 5, 8;
.0, 0, 0, 0, 0, 1, 10, 13;
.0, 0, 0, 0, 0, 0, 3, 20, 21;
.0, 0, 0, 0, 0, 0, 0, 9, 38, 34;
.0, 0, 0, 0, 0, 0, 0, 1, 22, 71, 55;
.0, 0, 0, 0, 0, 0, 0, 0, 4, 51, 130, 89;
.0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 111, 235, 144;

Cf. A037027.

T(n,k)=T(n-1,k-1)+T(n-2,k-2)+T(n-3,k-2), T(n,0)=0, T(1,1)=1, T(n,k)=0 if k<0 or if k>n . T(n,n)= Fibonacci(n)=A000045(n) . Sum_{k, 0<=k<=n}T(n,k)=A000073(n+1), tribonacci numbers . Sum_{n, n>=k}T(n,k)=A001045(k), Jacobsthal sequence.

A132964 Convolution triangle of A006190.

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 33, 29, 9, 1, 109, 126, 57, 12, 1, 360, 516, 306, 94, 15, 1, 1189, 2034, 1491, 600, 140, 18, 1, 3927, 7807, 6813, 3385, 1035, 195, 21, 1, 12970, 29382, 29737, 17568, 6630, 1638, 259, 24, 1, 42837, 108923, 125406, 85826, 38493, 11739, 2436, 332, 27, 1

Offset: 0

Philippe Deléham, Nov 24 2007

As a Riordan array, this is (1/(1-3x-x^2),x/(1-3x-x^2)).

T(n,k) is the number of words of length n over {0,1,2,3,4} having k letters 4 and avoiding runs of odd length for the letter 0. - Milan Janjic, Jan 14 2017

			Triangle begins:
      1;
      3,      1;
     10,      6,      1;
     33,     29,      9,     1;
    109,    126,     57,    12,     1;
    360,    516,    306,    94,    15,     1;
   1189,   2034,   1491,   600,   140,    18,    1;
   3927,   7807,   6813,  3385,  1035,   195,   21,   1;
  12970,  29382,  29737, 17568,  6630,  1638,  259,  24,  1;
  42837, 108923, 125406, 85826, 38493, 11739, 2436, 332, 27, 1;
  ...

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A006190, A037027, A054456, A112906.

Sum_{k=0..n} T(n,k) = A001076(n+1).
Sum_{k=0..floor(n/2)} T(n-k,k) = A007482(n).
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) + T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or k>n. - Philippe Deléham, Dec 08 2013

A181974 Triangle T(n,k), read by rows, given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -3, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 4, 2, 1, 5, 7, 5, 4, 1, 8, 11, 10, 9, 3, 1, 13, 18, 20, 20, 9, 5, 1, 21, 29, 38, 40, 22, 15, 4, 1, 34, 47, 71, 78, 51, 40, 14, 6, 1, 55, 76, 130, 147, 111, 95, 40, 22, 5, 1, 89, 123, 235, 272, 233, 213, 105, 68, 20, 7, 1

Offset: 0

Philippe Deléham, Apr 06 2012

			Triangle begins :
1
1, 1
2, 3, 1
3, 4, 2, 1
5, 7, 5, 4, 1
8, 11, 10, 9, 3, 1
13, 18, 20, 20, 9, 5, 1
21, 29, 38, 40, 22, 15, 4, 1
34, 47, 71, 78, 51, 40, 14, 6, 1
55, 76, 130, 147, 111, 95, 40, 22, 5, 1
89, 123, 235, 272, 233, 213, 105, 68, 20, 7, 1
144, 199, 420, 495, 474, 455, 256, 185, 65, 30, 6, 1

Cf. Columns : A000045, A000032, A001629, A023607, A001628, A152881, A001872, A001873

G.f.: (1+y*x+2*y*x^2)/(1-x-x^2-y^2*x^2).
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-2), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = 2, T(2,1) = 3 and T(n,k) = 0 if k<0 or if k>n.
T(n + 2k, 2k) = A037027(n + k, k).
T(n + 2k +1, 2k + 1) = A182001(n + k, k).
T(n,0) = Fibonacci(n+1).

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cp's OEIS Frontend

A238241 Riordan array (1/(1-x-x^2)^2, x/(1-x-x^2)^2).

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A054448 Triangle of partial row sums of triangle A054446(n,m), n >= m >= 0.

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A057281 Coefficient triangle of polynomials (falling powers) related to Fibonacci convolutions. Companion triangle to A057282.

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A091186 Triangle read by rows, in which n-th row gives expansion of x^n/((1-x)(1-x-x^2)^n).

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A160905 Right hand side of Pascal rhombus A059317.

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A202193 Triangle read by rows: T(n,m) = coefficient of x^n in expansion of (x/(1 - x - x^2 - x^3 - x^4))^m.

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A261507 Fibonacci-numbered rows of Pascal's triangle. Triangle read by rows: T(n,k)= binomial(Fibonacci(n), k).

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A123262 Fibonacci-tribonacci triangle.

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A132964 Convolution triangle of A006190.

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