A027934 -id:A027934 - cp's OEIS frontend

A055248 Triangle of partial row sums of triangle A007318(n,m) (Pascal's triangle). Triangle A008949 read backwards. Riordan (1/(1-2x), x/(1-x)).

1, 2, 1, 4, 3, 1, 8, 7, 4, 1, 16, 15, 11, 5, 1, 32, 31, 26, 16, 6, 1, 64, 63, 57, 42, 22, 7, 1, 128, 127, 120, 99, 64, 29, 8, 1, 256, 255, 247, 219, 163, 93, 37, 9, 1, 512, 511, 502, 466, 382, 256, 130, 46, 10, 1, 1024, 1023, 1013, 968, 848, 638, 386, 176, 56, 11, 1

Offset: 0

Wolfdieter Lang, May 26 2000

In the language of the Shapiro et al. reference (also given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The g.f. for the row polynomials p(n,x) (increasing powers of x) is 1/((1-2*z)*(1-x*z/(1-z))).
Binomial transform of the all 1's triangle: as a Riordan array, it factors to give (1/(1-x),x/(1-x))(1/(1-x),x). Viewed as a number square read by antidiagonals, it has T(n,k) = Sum_{j=0..n} binomial(n+k,n-j) and is then the binomial transform of the Whitney square A004070. - Paul Barry, Feb 03 2005
Riordan array (1/(1-2x), x/(1-x)). Antidiagonal sums are A027934(n+1), n >= 0. - Paul Barry, Jan 30 2005; edited by Wolfdieter Lang, Jan 09 2015
Eigensequence of the triangle = A005493: (1, 3, 10, 37, 151, 674, ...); row sums of triangles A011971 and A159573. - Gary W. Adamson, Apr 16 2009
Read as a square array, this is the generalized Riordan array ( 1/(1 - 2*x), 1/(1 - x) ) as defined in the Bala link (p. 5), which factorizes as ( 1/(1 - x), x/(1 - x) )*( 1/(1 - x), x )*( 1, 1 + x ) = P*U*transpose(P), where P denotes Pascal's triangle, A007318, and U is the lower unit triangular array with 1's on or below the main diagonal. - Peter Bala, Jan 13 2016

			The triangle a(n,m) begins:
n\m    0    1    2   3   4   5   6   7  8  9 10 ...
0:     1
1:     2    1
2:     4    3    1
3:     8    7    4   1
4:    16   15   11   5   1
5:    32   31   26  16   6   1
6:    64   63   57  42  22   7   1
7:   128  127  120  99  64  29   8   1
8:   256  255  247 219 163  93  37   9  1
9:   512  511  502 466 382 256 130  46 10  1
10: 1024 1023 1013 968 848 638 386 176 56 11  1
... Reformatted. - _Wolfdieter Lang_, Jan 09 2015
Fourth row polynomial (n=3): p(3,x)= 8 + 7*x + 4*x^2 + x^3.
The matrix inverse starts
   1;
  -2,   1;
   2,  -3,   1;
  -2,   5,  -4,    1;
   2,  -7,   9,   -5,    1;
  -2,   9, -16,   14,   -6,    1;
   2, -11,  25,-  30,   20,   -7,    1;
  -2,  13, -36,   55,  -50,   27,   -8,    1;
   2, -15,  49,  -91,  105,  -77,   35,   -9,  1;
  -2,  17, -64,  140, -196,  182, -112,   44, -10,   1;
   2, -19,  81, -204,  336, -378,  294, -156,  54, -11, 1;
   ...
which may be related to A029653. - _R. J. Mathar_, Mar 29 2013
From _Peter Bala_, Dec 23 2014: (Start)
With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins
/1      \ /1        \ /1       \       /1       \
|2 1     ||0 1       ||0 1      |      |2  1     |
|4 3 1   ||0 2 1     ||0 0 1    |... = |4  5 1   |
|8 7 4 1 ||0 4 3 1   ||0 0 2 1  |      |8 19 9 1 |
|...     ||0 8 7 4 1 ||0 0 4 3 1|      |...      |
|...     ||...       ||...      |      |         |
= A143494. (End)
Matrix factorization of square array as P*U*transpose(P):
/1      \ /1        \ /1 1 1 1 ...\    /1  1  1  1 ...\
|1 1     ||1 1       ||0 1 2 3 ... |   |2  3  4  5 ... |
|1 2 1   ||1 1 1     ||0 0 1 3 ... | = |4  7 11 16 ... |
|1 3 3 1 ||1 1 1 1   ||0 0 0 1 ... |   |8 15 26 42 ... |
|...     ||...       ||...         |   |...            |
- _Peter Bala_, Jan 13 2016

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Peter Bala, Notes on generalized Riordan arrays
Peter Bala, A055248: Rapidly converging series for log(2) and Pi
Jean-Luc Baril, Javier F. González, and José L. Ramírez, Last symbol distribution in pattern avoiding Catalan words, Univ. Bourgogne (France, 2022).
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Norman Lindquist and Gerard Sierksma, Extensions of set partitions, Journal of Combinatorial Theory, Series A 31.2 (1981): 190-198. See Table I.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

Column sequences: A000079 (powers of 2, m=0), A000225 (m=1), A000295 (m=2), A002662 (m=3), A002663 (m=4), A002664 (m=5), A035038 (m=6), A035039 (m=7), A035040 (m=8), A035041 (m=9), A035042 (m=10).
Row sums: A001792(n) = A055249(n, 0).
Alternating row sums: A011782.
Cf. A011971, A159573. - Gary W. Adamson, Apr 16 2009
Cf. A007318, A008949, A106516, A143494.

a055248 n k = a055248_tabl !! n !! k
a055248_row n = a055248_tabl !! n
a055248_tabl = map reverse a008949_tabl
-- Reinhard Zumkeller, Jun 20 2015

T := (n,k) -> 2^n - (1/2)*binomial(n, k-1)*hypergeom([1, n + 1], [n-k + 2], 1/2).
seq(seq(simplify(T(n,k)), k=0..n),n=0..10); # Peter Luschny, Oct 10 2019

a[n_, m_] := Sum[ Binomial[n, m + j], {j, 0, n}]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Paul Barry *)
T[n_, k_] := Binomial[n, k] * Hypergeometric2F1[1, k - n, k + 1, -1];
Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]]  (* Peter Luschny, Oct 06 2023 *)

a(n, m) = A008949(n, n-m), if n > m >= 0.
a(n, m) = Sum_{k=m..n} A007318(n, k) (partial row sums in columns m).
Column m recursion: a(n, m) = Sum_{j=m..n-1} a(j, m) + A007318(n, m) if n >= m >= 0, a(n, m) := 0 if nG.f. for column m: (1/(1-2*x))*(x/(1-x))^m, m >= 0.
a(n, m) = Sum_{j=0..n} binomial(n, m+j). - Paul Barry, Feb 03 2005
Inverse binomial transform (by columns) of A112626. - Ross La Haye, Dec 31 2006
T(2n,n) = A032443(n). - Philippe Deléham, Sep 16 2009
From Peter Bala, Dec 23 2014: (Start)
Exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(8 + 7*x + 4*x^2/2! + x^3/3!) = 8 + 15*x + 26*x^2/2! + 42*x^3/3! + 64*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ).
Let M denote the present triangle. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A143494 (but with a different offset). See the Example section. Cf. A106516. (End)
a(n,m) = Sum_{p=m..n} 2^(n-p)*binomial(p-1,m-1), n >= m >= 0, else 0. - Wolfdieter Lang, Jan 09 2015
T(n, k) = 2^n - (1/2)*binomial(n, k-1)*hypergeom([1, n+1], [n-k+2], 1/2). - Peter Luschny, Oct 10 2019
T(n, k) = binomial(n, k)*hypergeom([1, k - n], [k + 1], -1). - Peter Luschny, Oct 06 2023
n-th row polynomial R(n, x) = (2^n - x*(1 + x)^n)/(1 - x). These polynomials can be used to find series acceleration formulas for the constants log(2) and Pi. - Peter Bala, Mar 03 2025

A059570 Number of fixed points in all 231-avoiding involutions in S_n.

Original entry on oeis.org

1, 2, 6, 14, 34, 78, 178, 398, 882, 1934, 4210, 9102, 19570, 41870, 89202, 189326, 400498, 844686, 1776754, 3728270, 7806066, 16311182, 34020466, 70837134, 147266674, 305718158, 633805938, 1312351118, 2714180722, 5607318414, 11572550770, 23860929422

Offset: 1

Emeric Deutsch, Feb 16 2001

Number of odd parts in all compositions (ordered partitions) of n: a(3)=6 because in 3=2+1=1+2=1+1+1 we have 6 odd parts. Number of even parts in all compositions (ordered partitions) of n+1: a(3)=6 because in 4=3+1=1+3=2+2=2+1+1=1+2+1=1+1+2=1+1+1+1 we have 6 even parts.
Convolved with (1, 2, 2, 2, ...) = A001787: (1, 4, 12, 32, 80, ...). - Gary W. Adamson, May 23 2009
An elephant sequence, see A175654. For the corner squares 36 A[5] vectors, with decimal values between 15 and 480, lead to this sequence. For the central square these vectors lead to the companion sequence 4*A172481, for n>=-1. - Johannes W. Meijer, Aug 15 2010
a(n) is the total number of runs of equal parts in the compositions of n. a(5) = 34 because there are 34 runs of equal parts in the compositions of 5, with parentheses enclosing each run: (5), (4)(1), (1)(4), (3)(2), (2)(3), (3)(1,1), (1)(3)(1), (1,1)(3), (2,2)(1), (2)(1)(2), (1)(2,2), (2)(1,1,1), (1)(2)(1,1), (1,1)(2)(1), (1,1,1)(2), (1,1,1,1,1). - Gregory L. Simay, Apr 28 2017
a(n) - a(n-2) is the number of 1's in all compositions of n and more generally, the number of k's in all compositions of n+k-1. - Gregory L. Simay, May 01 2017

			a(3) = 6 because in the 231-avoiding involutions of {1,2,3}, i.e., in 123, 132, 213, 321, we have altogether 6 fixed points (3+1+1+1).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015.
S. Heubach and T. Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.
Brian Hopkins, Andrew V. Sills, Thotsaporn "Aek" Thanatipanonda, and Hua Wang, Parts and subword patterns in compositions, Preprint 2015.
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 30.
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Cf. A001787, A027934, A045623, A127984, A128255, A172481, A225084.

[(3*n+4)*2^n/18-2*(-1)^n/9: n in [1..40]]; // Vincenzo Librandi, May 01 2017

LinearRecurrence[{3,0,-4},{1,2,6},30] (* Harvey P. Dale, Dec 29 2013 *)
Table[(3 n + 4) 2^n/18 - 2 (-1)^n/9, {n, 30}] (* Vincenzo Librandi, May 01 2017 *)

a(n) = (3*n+4)*2^n/18 - 2*(-1)^n/9.
G.f.: z*(1-z)/((1+z)*(1-2*z)^2).
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(n-k, k+j)*2^k. - Paul Barry, Aug 29 2004
a(n) = Sum_{k=0..n+1} (-1)^(k+1)*binomial(n+1, k+j)*A001045(k). - Paul Barry, Jan 30 2005
Convolution of "Expansion of (1-x)/(1-x-2*x^2)" (A078008) with "Powers of 2" (A000079), treating the result as if offset=1. - Graeme McRae, Jul 12 2006
Convolution of "Difference sequence of A045623" (A045891) with "Positive integers repeated" (A008619), treating the result as if offset=1. - Graeme McRae, Jul 12 2006
a(n) = 3*a(n-1)-4*a(n-3); a(1)=1,a(2)=2,a(3)=6. - Philippe Deléham, Aug 30 2006
Equals row sums of A128255. (1, 2, 6, 14, 34, ...) - (0, 0, 1, 2, 6, 14, 34, ...) = A045623: (1, 2, 5, 12, 28, 64, ...). - Gary W. Adamson, Feb 20 2007
Equals triangle A059260 * [1, 2, 3, ...] as a vector. - Gary W. Adamson, Mar 06 2012
a(n) + a(n-1) = A001792(n-1). - Gregory L. Simay, Apr 30 2017
a(n) - a(n-2) = A045623(n-1). - Gregory L. Simay, May 01 2017
a(n) = A045623(n-1) + A045623(n-3) + A045623(n-5) + ... - Gregory L. Simay, Feb 19 2018
a(n) = A225084(2n,n). - Alois P. Heinz, Aug 30 2018

More terms from Eugene McDonnell (eemcd(AT)mac.com), Jan 13 2005

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

Original entry on oeis.org

1, 2, 6, 14, 36, 86, 210, 500, 1194, 2822, 6660, 15638, 36642, 85604, 199626, 464630, 1079892, 2506550, 5811762, 13462484, 31159914, 72071654, 166599972, 384912086, 888906306, 2052031172, 4735527306, 10925175254, 25198866036, 58108609526, 133973643090

Offset: 0

Johannes W. Meijer, Aug 06 2010; edited Jun 21 2013

a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the center square the bishop flies into a rage and turns into a raging elephant.
In chaturanga, the old Indian version of chess, one of the pieces was called gaja, elephant in Sanskrit. The Arabs called the game shatranj and the elephant became el fil in Arabic. In Spain chess became chess as we know it today but surprisingly in Spanish a bishop isn't a Christian bishop but a Moorish elephant and it still goes by its original name of el alfil.
On a 3 X 3 chessboard there are 2^9 = 512 ways for an elephant to fly into a rage on the central square (off the center the piece behaves like a normal bishop). The elephant is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 elephants lead to 46 different elephant sequences, see the overview of elephant sequences and the crossreferences.
The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the central square to A175655.

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 74, 366, 1992.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Viswanathan Anand, The Indian Defense, Time, Jun 19 2008.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Johannes W. Meijer, The elephant sequences.
Wikipedia, War Elephant.
Index entries for linear recurrences with constant coefficients, signature (3,1,-6).

Cf. Elephant sequences corner squares [decimal value A[5]]: A040000 [0], A000027 [16], A000045 [1], A094373 [2], A000079 [3], A083329 [42], A027934 [11], A172481 [7], A006138 [69], A000325 [26], A045623 [19], A000129 [21], A095121 [170], A074878 [43], A059570 [15], A175654 [71, this sequence], A026597 [325], A097813 [58], A057711 [27], 2*A094723 [23; n>=-1], A002605 [85], A175660 [171], A123203 [186], A066373 [59], A015518 [341], A134401 [187], A093833 [343].

Cf. A000244, A006130, A006138, A075188, A175655, A180140.

[n le 3 select Factorial(n) else 3*Self(n-1) +Self(n-2) -6*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 08 2021

nmax:=28; m:=1; A[1]:=[0,0,0,0,1,0,0,0,1]: A[2]:=[0,0,0,1,0,1,0,0,0]: A[3]:=[0,0,0,0,1,0,1,0,0]: A[4]:=[0,1,0,0,0,0,0,1,0]: A[5]:=[0,0,1,0,0,0,1,1,1]: A[6]:=[0,1,0,0,0,0,0,1,0]: A[7]:=[0,0,1,0,1,0,0,0,0]: A[8]:=[0,0,0,1,0,1,0,0,0]: A[9]:=[1,0,0,0,1,0,0,0,0]: A:=Matrix([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{3,1,-6}, {1,2,6}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;2;6])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[( (1-x-x^2)/((1-2*x)*(1-x-3*x^2)) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Dec 08 2021

G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).
a(n) = 3*a(n-1) + a(n-2) - 6*a(n-3) with a(0)=1, a(1)=2 and a(2)=6.
a(n) = ((6+10*A)*A^(-n-1) + (6+10*B)*B^(-n-1))/13 - 2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6.
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n) - A006130(n-1)*sqrt(13)).
a(n) = b(n) - b(n-1) - b(n-2), where b(n) = Sum_{k=1..n} Sum_{j=0..k} binomial(j,n-3*k+2*j)*(-6)^(k-j)*binomial(k,j)*3^(3*k-n-j), n>0, b(0)=1, with a(0) = b(0), a(1) = b(1) - b(0). - Vladimir Kruchinin, Aug 20 2010
a(n) = 2*A006138(n) - 2^n = 2*(A006130(n) + A006130(n-1)) - 2^n. - G. C. Greubel, Dec 08 2021
E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 3*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - Stefano Spezia, Feb 12 2023

A099036 a(n) = 2^n - Fibonacci(n).

Original entry on oeis.org

1, 1, 3, 6, 13, 27, 56, 115, 235, 478, 969, 1959, 3952, 7959, 16007, 32158, 64549, 129475, 259560, 520107, 1041811, 2086206, 4176593, 8359951, 16730848, 33479407, 66987471, 134021310, 268117645, 536356683, 1072909784, 2146137379, 4292788987, 8586410014

Offset: 0

Paul Barry, Sep 23 2004

Binomial transform of (-1)^n*A000045(n) + 1 = (-1)^n*A008346(n).
Number of compositions of n+1 that contain 1 as a part. - Vladeta Jovovic, Sep 26 2004
Generated from iterates of M * [1,1,1,...], where M = a tridiagonal matrix with [0,1,1,1,...] as the main diagonal, [1,1,1,...] as the superdiagonal and [1,0,0,0,...] as the subdiagonal. - Gary W. Adamson, Jan 05 2009
Starting with offset 1, generated from iterates of M * [1,1,1,...], M*ANS -> M*ANS,...; where M = = a tridiagonal matrix with (0,1,1,1,...) in the main diagonal, (1,1,1,...) in the superdiagonal and (1,0,0,0,...) in the subdiagonal. - Gary W. Adamson, Jan 04 2009
An elephant sequence, see A175655. For the central square 24 A[5] vectors, with decimal values between 11 and 416, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A027934 (without the leading 0). - Johannes W. Meijer, Aug 15 2010
Number of fixed points in all compositions of n+1. - Alois P. Heinz, Jun 18 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Cf. A000045, A000079, A003056, A008346, A027934, A101220, A117591, A175655, A212323, A238350.

a099036 n = a099036_list !! n
a099036_list = zipWith (-) a000079_list a000045_list
-- Reinhard Zumkeller, Aug 15 2013

[2^n-Fibonacci(n): n in [0..35]]; // Vincenzo Librandi, May 03 2011

Table[2^n-Fibonacci[n],{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)

a(n)=2^n-fibonacci(n) \\ Charles R Greathouse IV, Sep 24 2015

def A099036(n): return 2**n -fibonacci(n) # G. C. Greubel, Jun 05 2025

G.f.: (1 - x)^2/((1 - 2*x)*(1 - x - x^2)).
a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3).
a(n) = A101220(1,2,n+1) - A101220(1,2,n). - Ross La Haye, Aug 05 2005
a(n) = A000079(n+1) - A117591(n) = A117591(n) - 2 * A000045(n). - Reinhard Zumkeller, Aug 15 2013
a(n) = Sum_{t_1+2*t_2+...+n*t_n = n} multinomial(1+t_1+t_2+...+t_n, 1+t_1, t_2, ..., t_n). - Mircea Merca, Oct 09 2013
a(n) = Sum_{k=0..A003056(n+1)} k * A238350(n+1,k). - Alois P. Heinz, Jun 18 2020
E.g.f.: cosh(2*x) + sinh(2*x) - 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Jan 31 2023

More terms from Ross La Haye, Aug 05 2005

A105809 Riordan array (1/(1 - x - x^2), x/(1 - x)).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 5, 7, 7, 4, 1, 8, 12, 14, 11, 5, 1, 13, 20, 26, 25, 16, 6, 1, 21, 33, 46, 51, 41, 22, 7, 1, 34, 54, 79, 97, 92, 63, 29, 8, 1, 55, 88, 133, 176, 189, 155, 92, 37, 9, 1, 89, 143, 221, 309, 365, 344, 247, 129, 46, 10, 1, 144, 232, 364, 530, 674, 709, 591

Offset: 0

Paul Barry, May 04 2005

Previous name was: A Fibonacci-Pascal matrix.
From Wolfdieter Lang, Oct 04 2014: (Start)
In the column k of this triangle (without leading zeros) is the k-fold iterated partial sums of the Fibonacci numbers, starting with 1. A000045(n+1), A000071(n+3), A001924(n+1), A014162(n+1), A014166(n+1), ..., n >= 0. See the Riordan property.
For a combinatorial interpretation of these iterated partial sums see the H. Belbachir and A. Belkhir link. There table 1 shows in the rows these columns. In their notation (with r = k) f^(k)(n) = T(k, n+k).
The A-sequence of this Riordan triangle is [1, 1] (see the recurrence for T(n, k), k >= 1, given in the formula section). The Z-sequence is A165326 = [1, repeat(1, -1)]. See the W. Lang link under A006232 for Riordan A- and Z-sequences. (End)

			The triangle T(n,k) begins:
n\k   0   1   2    3    4    5    6    7    8   9  10 11 12 13 ...
0:    1
1:    1   1
2:    2   2   1
3:    3   4   3    1
4:    5   7   7    4    1
5:    8  12  14   11    5    1
6:   13  20  26   25   16    6    1
7:   21  33  46   51   41   22    7    1
8:   34  54  79   97   92   63   29    8    1
9:   55  88 133  176  189  155   92   37    9   1
10:  89 143 221  309  365  344  247  129   46  10   1
11: 144 232 364  530  674  709  591  376  175  56  11  1
12: 233 376 596  894 1204 1383 1300  967  551 231  67 12  1
13: 377 609 972 1490 2098 2587 2683 2267 1518 782 298 79 13  1
... reformatted and extended - _Wolfdieter Lang_, Oct 03 2014
------------------------------------------------------------------
Recurrence from Z-sequence (see a comment above): 8 = T(0,5) = (+1)*5 + (+1)*7 + (-1)*7 + (+1)*4 + (-1)*1 = 8. - _Wolfdieter Lang_, Oct 04 2014

Reinhard Zumkeller, Rows n = 0..120 of table, flattened
H. Belbachir and A. Belkhir, Combinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.4.3.
Hung Viet Chu, Partial Sums of the Fibonacci Sequence, arXiv:2106.03659 [math.CO], 2021.
Anton Vladimirovich Shutov, On some analogue of the Gelfond problem for Zeckendorf representations, Chebyshevskii Sbornik (2024) Vol. 25, No. 5, 195-215. See p. 215. (In Russian)
Index entries for triangles and arrays related to Pascal's triangle

Cf. A165326 (Z-sequence), A027934 (row sums), A010049(n+1) (antidiagonal sums), A212804 (alternating row sums), inverse is A105810.

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A109906, A111006, A114197, A162741, A228074.

a105809 n k = a105809_tabl !! n !! k
a105809_row n = a105809_tabl !! n
a105809_tabl = map fst $ iterate
   (\(u:_, vs) -> (vs, zipWith (+) ([u] ++ vs) (vs ++ [0]))) ([1], [1,1])
-- Reinhard Zumkeller, Aug 15 2013

T := (n,k) -> `if`(n=0,1,binomial(n,k)*hypergeom([1,k/2-n/2,k/2-n/2+1/2], [k+1,-n], -4)); for n from 0 to 13 do seq(simplify(T(n,k)),k=0..n) od; # Peter Luschny, Oct 10 2014

T[n_, k_] := Sum[Binomial[n-j, k+j], {j, 0, n}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] (* Jean-François Alcover, Jun 11 2019 *)

Riordan array (1/(1-x-x^2), x/(1-x)).
Triangle T(n, k) = Sum_{j=0..n} binomial(n-j, k+j); T(n, 0) = A000045(n+1);
T(n, m) = T(n-1, m-1) + T(n-1, m).
T(n, k) = Sum_{j=0..n} binomial(j, n+k-j). - Paul Barry, Oct 23 2006
G.f. of row polynomials Sum_{k=0..n} T(n, k)*x^k is (1 - z)/((1 - z - z^2)*(1 - (1 + x)*z)) (Riordan property). - Wolfdieter Lang, Oct 04 2014
T(n, k) = binomial(n, k)*hypergeom([1, k/2 - n/2, k/2 - n/2 + 1/2],[k + 1, -n], -4) for n > 0. - Peter Luschny, Oct 10 2014
From Wolfdieter Lang, Feb 13 2025: (Start)
Array A(k, n) = Sum_{j=0..n} F(j+1)*binomial(k-1+n-j, k-1), k >= 0, n >= 0, with F = A000045, (from Riordan triangle k-th convolution in columns without leading 0s).
A(k, n) = F(n+1+2*k) - Sum_{j=0..k-1} F(2*(k-j)-1) * binomial(n+1+j, j), (from iteration of partial sums).
Triangle T(n, k) = A(k, n-k) = Sum_{j=k..n} F(n-j+1) * binomial(j-1, k-1), 0 <= k <= n.
T(n, k) = F(n+1+k) - Sum_{j=0..k-1} F(2*(k-j)-1) * binomial(n - (k-1-j), j). (End)
T(n, k) =  A027926(n, n+k), for 0 <= k <= n. - Wolfdieter Lang, Mar 08 2025

Use first formula as a more descriptive name, Joerg Arndt, Jun 08 2021

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

Original entry on oeis.org

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

A108617 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) for 0 < k < n, T(n,0) = T(n,n) = n-th Fibonacci number.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 3, 2, 3, 5, 6, 5, 3, 5, 8, 11, 11, 8, 5, 8, 13, 19, 22, 19, 13, 8, 13, 21, 32, 41, 41, 32, 21, 13, 21, 34, 53, 73, 82, 73, 53, 34, 21, 34, 55, 87, 126, 155, 155, 126, 87, 55, 34, 55, 89, 142, 213, 281, 310, 281, 213, 142, 89, 55, 89, 144, 231, 355, 494, 591, 591, 494, 355, 231, 144, 89

Offset: 0

Reinhard Zumkeller, Jun 12 2005

Sum of n-th row = 2*A027934(n). - Reinhard Zumkeller, Oct 07 2012

			Triangle begins:
   0;
   1,   1;
   1,   2,   1;
   2,   3,   3,   2;
   3,   5,   6,   5,   3;
   5,   8,  11,  11,   8,   5;
   8,  13,  19,  22,  19,  13,   8;
  13,  21,  32,  41,  41,  32,  21,  13;
  21,  34,  53,  73,  82,  73,  53,  34,  21;
  34,  55,  87, 126, 155, 155, 126,  87,  55,  34;
  55,  89, 142, 213, 281, 310, 281, 213, 142,  89,  55;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Hacéne Belbachir and László Szalay, On the Arithmetic Triangles, Šiauliai Mathematical Seminar, Vol. 9 (17), 2014. See Fig. 1 p. 18.
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Pascal's Triangle.
Wikipedia, Fibonacci number.
Wikipedia, Pascal's triangle.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A074829, A108037.

T(2n,n) gives 2*A176085(n).

a108617 n k = a108617_tabl !! n !! k
a108617_row n = a108617_tabl !! n
a108617_tabl = [0] : iterate f [1,1] where
   f row@(u:v:_) = zipWith (+) ([v - u] ++ row) (row ++ [v - u])
-- Reinhard Zumkeller, Oct 07 2012

function T(n,k) // T = A108617
  if k eq 0 or k eq n then return Fibonacci(n);
  else return T(n-1,k-1) + T(n-1,k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 20 2023

A108617 := proc(n,k) option remember;
    if k = 0 or k=n then
        combinat[fibonacci](n) ;
    elif k <0 or k > n then
        0 ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Oct 05 2012

a[1]:={0}; a[n_]:= a[n]= Join[{Fibonacci[#]}, Map[Total, Partition[a[#],2,1]], {Fibonacci[#]}]&[n-1]; Flatten[Map[a, Range[15]]] (* Peter J. C. Moses, Apr 11 2013 *)

def T(n,k): # T = A108617
    if (k==0 or k==n): return fibonacci(n)
    else: return T(n-1,k-1) + T(n-1,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 20 2023

T(n,0) = T(n,n) = A000045(n);
T(n,1) = T(n,n-1) = A000045(n+1) for n>0;
T(n,2) = T(n,n-2) = A000045(n+2) - 2 = A001911(n-1) for n>1;
Sum_{k=0..n} T(n,k) = 2*A027934(n-1) for n>0.
Sum_{k=0..n} (-1)^k*T(n, k) = 2*((n+1 mod 2)*Fibonacci(n-2) + [n=0]). - G. C. Greubel, Oct 20 2023

A167821 a(n) is the number of n-tosses having a run of 3 or more heads or a run of 3 or more tails for a fair coin (i.e., probability is a(n)/2^n).

Original entry on oeis.org

0, 0, 2, 6, 16, 38, 86, 188, 402, 846, 1760, 3630, 7438, 15164, 30794, 62342, 125904, 253782, 510758, 1026684, 2061730, 4136990, 8295872, 16627166, 33311646, 66716028, 133582106, 267406998, 535206832, 1071049286, 2143127030, 4287918140, 8578528818

Offset: 1

V.J. Pohjola, Nov 13 2009

A167821(n) is the difference between A000918(n), the number of branches of a complete binary tree of n levels, and the number of recursive calls needed to compute the (n+1)-th Fibonacci number F(n+1) as defined in A019274: A167821(n) = A000918(n) - A019274(n+1). - Denis Lorrain, Jan 14 2012

Partial sums of A027934 multiplied term by term by 2 (as shown by the second formula), i.e., partial sums of row sums of A108617. - J. M. Bergot, Oct 02 2012, clarified by R. J. Mathar, Oct 05 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3, -1, -2).

Cf. A008466, A050231.

[2^n-2*Fibonacci(n+1): n in [1..40]]; // Vincenzo Librandi, Jun 28 2016

CoefficientList[Series[(2 x^2)/(1 - 3 x + x^2 + 2 x^3), {x, 0, 30}], x]
Table[2^n - 2*Fibonacci[n + 1], {n, 1, 31}]
LinearRecurrence[{3, -1, -2}, {0, 0, 2}, 50] (* G. C. Greubel, Jun 27 2016 *)

G.f.: (2 x^2)/(1 - 3 x + x^2 + 2 x^3);
a(n) = 2^n - 2*Fibonacci(n+1).
a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3). - G. C. Greubel, Jun 27 2016

Showing 1-10 of 20 results. Next

cp's OEIS Frontend

A192744 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A101220 a(n) = Sum_{k=0..n} Fibonacci(n-k)*n^k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A055248 Triangle of partial row sums of triangle A007318(n,m) (Pascal's triangle). Triangle A008949 read backwards. Riordan (1/(1-2x), x/(1-x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A059570 Number of fixed points in all 231-avoiding involutions in S_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A099036 a(n) = 2^n - Fibonacci(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).