A039717 -id:A039717 - cp's OEIS frontend

A030528 Triangle read by rows: a(n,k) = binomial(k,n-k).

1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 3, 4, 1, 0, 0, 1, 6, 5, 1, 0, 0, 0, 4, 10, 6, 1, 0, 0, 0, 1, 10, 15, 7, 1, 0, 0, 0, 0, 5, 20, 21, 8, 1, 0, 0, 0, 0, 1, 15, 35, 28, 9, 1, 0, 0, 0, 0, 0, 6, 35, 56, 36, 10, 1, 0, 0, 0, 0, 0, 1, 21, 70, 84, 45, 11, 1, 0, 0, 0, 0, 0, 0, 7, 56, 126, 120, 55, 12, 1

Offset: 1

Wolfdieter Lang

A convolution triangle of numbers obtained from A019590.
a(n,m) := s1(-1; n,m), a member of a sequence of triangles including s1(0; n,m)= A023531(n,m) (unit matrix) and s1(2; n,m)= A007318(n-1,m-1) (Pascal's triangle).
The signed triangular matrix a(n,m)*(-1)^(n-m) is the inverse matrix of the triangular Catalan convolution matrix A033184(n+1,m+1), n >= m >= 0, with A033184(n,m) := 0 if nRiordan array (1+x, x(1+x)). The signed triangle is the Riordan array (1-x,x(1-x)), inverse to (c(x),xc(x)) with c(x) g.f. for A000108. - Paul Barry, Feb 02 2005 [with offset 0]
Also, a(n,k)=number of compositions of n into k parts of 1's and 2's. Example: a(6,4)=6 because we have 2211, 2121, 2112, 1221, 1212 and 1122. - Emeric Deutsch, Apr 05 2005 [see MacMahon and Riordan. - Wolfdieter Lang, Jul 27 2023]
Subtriangle of A026729. - Philippe Deléham, Aug 31 2006
a(n,k) is the number of length n-1 binary sequences having no two consecutive 0's with exactly k-1 1's. Example: a(6,4)=6 because we have 01011, 01101, 01110, 10101, 10110, 11010. - Geoffrey Critzer, Jul 22 2013
Mirrored, shifted Fibonacci polynomials of A011973. The polynomials (illustrated below) of this entry have the property that p(n,t) = t * [p(n-1,t) + p(n-2,t)]. The additive properties of Pascal's triangle (A007318) are reflected in those of these polynomials, as can be seen in the Example Section below and also when the o.g.f. G(x,t) below is expanded as the series x*(1+x) + t * [x*(1+x)]^2 + t^2 * [x*(1+x)]^3 + ... . See also A053122 for a relation to Cartan matrices. - Tom Copeland, Nov 04 2014
Rows of this entry appear as columns of an array for an infinitesimal generator presented in the Copeland link. - Tom Copeland, Dec 23 2015
For n >= 2, the n-th row is also the coefficients of the vertex cover polynomial of the (n-1)-path graph P_{n-1}. - Eric W. Weisstein, Apr 10 2017
With an additional initial matrix element a_(0,0) = 1 and column of zeros a_(n,0) = 0 for n > 0, these are antidiagonals read from bottom to top of the numerical coefficients of the Maurer-Cartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper, which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Cf. A011973. And A169803. - Tom Copeland, Jul 02 2018

			Triangle starts:
  [ 1]  1
  [ 2]  1   1
  [ 3]  0   2   1
  [ 4]  0   1   3   1
  [ 5]  0   0   3   4   1
  [ 6]  0   0   1   6   5   1
  [ 7]  0   0   0   4  10   6   1
  [ 8]  0   0   0   1  10  15   7   1
  [ 9]  0   0   0   0   5  20  21   8   1
  [10]  0   0   0   0   1  15  35  28   9   1
  [11]  0   0   0   0   0   6  35  56  36  10   1
  [12]  0   0   0   0   0   1  21  70  84  45  11   1
  [13]  0   0   0   0   0   0   7  56 126 120  55  12   1
  ...
From _Tom Copeland_, Nov 04 2014: (Start)
For quick comparison to other polynomials:
  p(1,t) = 1
  p(2,t) = 1 + 1 t
  p(3,t) = 0 + 2 t + 1 t^2
  p(4,t) = 0 + 1 t + 3 t^2 + 1 t^3
  p(5,t) = 0 + 0   + 3 t^2 + 4 t^3 +  1 t^4
  p(6,t) = 0 + 0   + 1 t^2 + 6 t^3 +  5 t^4 +  1 t^5
  p(7,t) = 0 + 0   + 0     + 4 t^3 + 10 t^4 +  6 t^5 + 1 t^6
  p(8,t) = 0 + 0   + 0     + 1 t^3 + 10 t^4 + 15 t^5 + 7 t^6 + 1 t^7
  ...
Reading along columns gives rows for Pascal's triangle. (End)

P. A. MacMahon, Combinatory Analysis, Two volumes (bound as one), Chelsea Publishing Company, New York, 1960, Vol. I, nr. 124, p. 151.
John Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, London, 1958. eq. (35), p.124, 11. p. 154.

Indranil Ghosh, Rows 1..125 of triangle, flattened
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 18.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, Addendum to Elliptic Lie Triad
Loïc Foissy, Cointeraction on noncrossing partitions and related polynomial invariants, arXiv:2501.18212 [math.CO], 2025. See p. 21.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Donatella Merlini, Renzo Sprugnoli, and M. Cecilia Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000.
Donatella Merlini, Renzo Sprugnoli, and M. Cecilia Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139.
Peter J. Olver, The canonical contact form.
Eric Weisstein's World of Mathematics, Path Graph
Eric Weisstein's World of Mathematics, Vertex Cover Polynomial

Row sums A000045(n+1) (Fibonacci). a(n, 1)= A019590(n) (Fermat's last theorem). Cf. A049403.

Cf. A104597, A146559, A155020, A125145, A057078, A039717, A001792, A057862, A011973, A115139.

/* As triangle */ [[Binomial(k, n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Nov 05 2014

for n from 1 to 12 do seq(binomial(k,n-k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Apr 05 2005

nn=10;CoefficientList[Series[(1+x)/(1-y x - y x^2),{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Jul 22 2013 *)
Table[Binomial[k, n - k], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Dec 23 2015 *)
CoefficientList[Table[x^(n/2 - 1) Fibonacci[n + 1, Sqrt[x]], {n, 10}],
   x] // Flatten (* Eric W. Weisstein, Apr 10 2017 *)

a(n, m) = 2*(2*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

G.f. for m-th column: (x*(1+x))^m.

As a number triangle with offset 0, this is T(n, k) = Sum_{i=0..n} (-1)^(n+i)*binomial(n, i)*binomial(i+k+1, 2k+1). The antidiagonal sums give the Padovan sequence A000931(n+5). Inverse binomial transform of A078812 (product of lower triangular matrices). - Paul Barry, Jun 21 2004

G.f.: (1 + x)/(1 - y*x - y*x^2). - Geoffrey Critzer, Jul 22 2013 [offset 0] [with offset 1: g.f. of row polynomials in y: x*(1+x)*y/(1 - x*(1+x)*y). - Wolfdieter Lang, Jul 27 2023]

From Tom Copeland, Nov 04 2014: (Start)

O.g.f: G(x,t) = x*(1+x) / [1 - t*x*(1+x)] = -P[Cinv(-x),t], where P(x,t)= x / (1 + t*x) and Cinv(x)= x*(1-x) are the compositional inverses in x of Pinv(x,t) = -P(-x,t) = x / (1 - t*x) and C(x) = [1-sqrt(1-4*x)]/2, an o.g.f. for the shifted Catalan numbers A000108.

Therefore, Ginv(x,t) = -C[Pinv(-x,t)] = {-1 + sqrt[1 + 4*x/(1+t*x)]}/2, which is -A124644(-x,t).

This places this array in a family of arrays related by composition of P and C and their inverses and interpolation by t, such as A091867 and A104597, and associated to the Catalan, Motzkin, Fine, and Fibonacci numbers. Cf. A104597 (polynomials shifted in t) A125145, A146559, A057078, A000045, A155020, A125145, A039717, A001792, A057862, A011973, A115139. (End)

More terms from Emeric Deutsch, Apr 05 2005

A030191 Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2.

Original entry on oeis.org

1, 5, 20, 75, 275, 1000, 3625, 13125, 47500, 171875, 621875, 2250000, 8140625, 29453125, 106562500, 385546875, 1394921875, 5046875000, 18259765625, 66064453125, 239023437500, 864794921875, 3128857421875, 11320312500000, 40957275390625, 148184814453125

Offset: 0

Wolfdieter Lang

Number of (s(0), s(1), ..., s(2n+4)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+4, s(0) = 1, s(2n+4) = 5. - Herbert Kociemba, Jun 14 2004
Binomial transform of A002878. - Philippe Deléham, Oct 04 2005
Diagonal of square array A216219. - Philippe Deléham, Mar 15 2013
Lim_{n->oo} a(n+1)/a(n) = 2 + phi = A296184, where phi = A001622. - Wolfdieter Lang, Nov 16 2023~

			G.f. = 1 + 5*x + 20*x^2 + 75*x^3 + 275*x^4 + 1000*x^5 + 3625*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Santiago Alzate, Oscar Correa, and Rigoberto Flórez, Fibonacci identities from Jordan Identities, arXiv:2009.02639 [math.NT], 2020.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0 ,b=1; p=5, q=-5.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (38) and (45), lhs, m=5.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (5,-5).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000032, A000045, A001622.

Cf. A002878, A039717, A081567, A109466, A216219.

a:=[1,5];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # G. C. Greubel, Dec 28 2019

I:=[1,5]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // G. C. Greubel, Dec 28 2019

seq(coeff(series(1/(1-5*x+5*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 28 2019

Table[MatrixPower[{{2,1},{1,3}},n][[1]][[2]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
a[ n_]:= (((5 + Sqrt[5])/2)^(n + 1) - ((5 - Sqrt[5])/2)^(n + 1)) / Sqrt[5] // Expand; (* Michael Somos, Aug 27 2015 *)
Table[If[EvenQ[n], 5^(n/2)*LucasL[n+1], 5^((n+1)/2)*Fibonacci[n+1]], {n,0,30}] (* G. C. Greubel, Dec 28 2019 *)

{a(n) = imag((quadgen(5) + 2)^(n+1))}; /* Michael Somos, Aug 27 2015 */

[lucas_number1(n,5,5) for n in range(1, 22)] # Zerinvary Lajos, Apr 22 2009

a(n) = (sqrt(5))^n*U(n, sqrt(5)/2).
G.f.: 1/(1 - 5*x + 5*x^2).
a(2*k+1) = 5^(k+1)*Fibonacci(2*k+2).
a(2*k) = 5^k*Lucas(2*k+1).
a(n-1) = Sum_{k=0..n} C(n, k)*Fibonacci(2*k). - Benoit Cloitre, Jun 21 2003
a(n) = 5*a(n-1) - 5*a(n-2). - Benoit Cloitre, Oct 23 2003
a(n-1) = (((5+sqrt(5))/2)^n - ((5-sqrt(5))/2)^n)/sqrt(5) is the 2nd binomial transform of Fibonacci(n), the first binomial transform of Fibonacci(2n) and its n-th term is the n-th term of the third binomial transform of Fibonacci(3n) divided by 2^n. - Paul Barry, Mar 23 2004
a(n) = Sum_{k-0..n} 5^k*A109466(n,k). - Philippe Deléham, Nov 28 2006
a(n) = 5*A039717(n), n>0. - Philippe Deléham, Mar 12 2013
a(n) = A216219(n,n+3) = A216219(n,n+4) = A216219(n+3,n) = A216219(n+4,n). - Philippe Deléham, Mar 15 2013
G.f.: 1/(1-5*x/(1+x/(1-x))). - Philippe Deléham, Mar 15 2013
a(n) = -a(-2-n) * 5^(n+1) for all n in Z. - Michael Somos, Aug 27 2015
E.g.f.: exp((5-sqrt(5))*x/2)*((5 + sqrt(5))*exp(sqrt(5)*x) - 5 + sqrt(5))/(2*sqrt(5)). - Stefano Spezia, Dec 29 2019
a(n) = Sum_{k=0..n} A081567(n-k)*2^k. - Philippe Deléham, Mar 10 2023

A081567 Second binomial transform of F(n+1).

Original entry on oeis.org

1, 3, 10, 35, 125, 450, 1625, 5875, 21250, 76875, 278125, 1006250, 3640625, 13171875, 47656250, 172421875, 623828125, 2257031250, 8166015625, 29544921875, 106894531250, 386748046875, 1399267578125, 5062597656250, 18316650390625, 66270263671875, 239768066406250

Offset: 0

Paul Barry, Mar 22 2003

Binomial transform of F(2*n-1), index shifted by 1, where F is A000045. - corrected by Richard R. Forberg, Aug 12 2013
Case k=2 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=1, a(1)=k+1.
Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1, 2, ..., 2*n+1, s(0) = 3, s(2*n+1) = 4.
a(n+1) gives the number of periodic multiplex juggling sequences of length n with base state <2>. - Steve Butler, Jan 21 2008
a(n) is also the number of idempotent order-preserving partial transformations (of an n-element chain) of waist n (waist(alpha) = max(Im(alpha))). - Abdullahi Umar, Sep 14 2008
Counts all paths of length (2*n+1), n>=0, starting at the initial node on the path graph P_9, see the Maple program. - Johannes W. Meijer, May 29 2010
Given the 3 X 3 matrix M = [1,1,1; 1,1,0; 1,1,3], a(n) = term (1,1) in M^(n+1). - Gary W. Adamson, Aug 06 2010
Number of nonisomorphic graded posets with 0 and 1 of rank n+2, with exactly 2 elements of each rank level between 0 and 1. Also the number of nonisomorphic graded posets with 0 of rank n+1, with exactly 2 elements of each rank level above 0. (This is by Stanley's definition of graded, that all maximal chains have the same length.) - David Nacin, Feb 26 2012
a(n) = 3^n a(n;1/3) = Sum_{k=0..n} C(n,k) * F(k-1) * (-1)^k * 3^(n-k), which also implies the Deleham formula given below and where a(n;d), n=0,1,...,d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also the papers of Witula et al.). - Roman Witula, Jul 12 2012
The limiting ratio a(n)/a(n-1) is 1 + phi^2. - Bob Selcoe, Mar 17 2014
a(n) counts closed walks on K_2 containing 3 loops on the index vertex and 2 loops on the other. Equivalently the (1,1) entry of A^n where the adjacency matrix of digraph is A=(3,1; 1,2). - David Neil McGrath, Nov 18 2014

			a(4)=125: 35*(3 + (35 mod 10 - 10 mod 3)/(10-3)) = 35*(3 + 4/7) = 125. - _Bob Selcoe_, Mar 17 2014

R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pages 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Santiago Alzate, Oscar Correa, and Rigoberto Flórez, Fibonacci identities from Jordan Identities, arXiv:2009.02639 [math.NT], 2020.
Carolina Benedetti, Christopher R. H. Hanusa, Pamela E. Harris, Alejandro H. Morales, and Anthony Simpson, Kostant's partition function and magic multiplex juggling sequences, arXiv:2001.03219 [math.CO], 2020. See Table 1 p. 12.
S. Butler and R. Graham, Enumerating (multiplex) juggling sequences, arXiv:0801.2597 [math.CO], 2008.
P. E. Harris, E. Insko, and L. K. Williams, The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula, arXiv preprint arXiv:1401.0055 [math.RT], 2013.
Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, Binomials Transformation Formulae of Scaled Fibonacci Numbers, Open Mathematics, 15(1) (2017), 477-485.
A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), #04.3.8.
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
D. Nacin, The Minimal Non-Koszul A(Gamma), arXiv preprint arXiv:1204.1534 [math.QA], 2012. - From _N. J. A. Sloane_, Oct 05 2012
Roman Witula and Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310-329, MR2555042.
Index entries for linear recurrences with constant coefficients, signature (5,-5).

a(n) = 5*A052936(n-1), n > 1.
Row sums of A114164.
Cf. A000045, A007051 (INVERTi transform), A007598, A028387, A030191, A039717, A049310, A081568 (binomial transform), A086351 (INVERT transform), A090041, A093129, A094441, A111776, A147748, A178381, A189315.

I:=[1, 3]; [n le 2 select I[n] else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 27 2012

with(GraphTheory):G:=PathGraph(9): A:= AdjacencyMatrix(G): nmax:=23; n2:=nmax*2+2: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..9); od: seq(a(2*n+1),n=0..nmax); # Johannes W. Meijer, May 29 2010

Table[MatrixPower[{{2,1},{1,3}},n][[2]][[2]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
LinearRecurrence[{5,-5},{1,3},30] (* Vincenzo Librandi, Feb 27 2012 *)

Vec((1-2*x)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Mar 18 2014

def a(n, adict={0:1, 1:3}):
    if n in adict:
        return adict[n]
    adict[n]=5*a(n-1) - 5*a(n-2)
    return adict[n] # David Nacin, Mar 04 2012

a(n) = 5*a(n-1) - 5*a(n-2) for n >= 2, with a(0) = 1 and a(1) = 3.
a(n) = (1/2 - sqrt(5)/10) * (5/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2) * (sqrt(5)/2 + 5/2)^n.
G.f.: (1 - 2*x)/(1 - 5*x + 5*x^2).
a(n-1) = Sum_{k=1..n} binomial(n, k)*F(k)^2. - Benoit Cloitre, Oct 26 2003
a(n) = A090041(n)/2^n. - Paul Barry, Mar 23 2004
The sequence 0, 1, 3, 10, ... with a(n) = (5/2 - sqrt(5)/2)^n/5 + (5/2 + sqrt(5)/2)^n/5 - 2(0)^n/5 is the binomial transform of F(n)^2 (A007598). - Paul Barry, Apr 27 2004
From Paul Barry, Nov 15 2005: (Start)
a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(n, j)*binomial(j+k, 2k);
a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(n, k+j)*binomial(k, k-j)2^(n-k-j);
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} binomial(n+k-j, n-k-j)*binomial(k, j)(-1)^j*2^(n-k-j). (End)
a(n) = A111776(n, n). - Abdullahi Umar, Sep 14 2008
a(n) = Sum_{k=0..n} A094441(n,k)*2^k. - Philippe Deléham, Dec 14 2009
a(n+1) = Sum_{k=-floor(n/5)..floor(n/5)} ((-1)^k*binomial(2*n, n+5*k)/2). -Mircea Merca, Jan 28 2012
a(n) = A030191(n) - 2*A030191(n-1). - R. J. Mathar, Jul 19 2012
G.f.: Q(0,u)/x - 1/x, where u=x/(1-2*x), Q(k,u) = 1 + u^2 + (k+2)*u - u*(k+1 + u)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
For n>=3: a(n) = a(n-1)*(3+(a(n-1) mod a(n-2) - a(n-2) mod a(n-3))/(a(n-2) - a(n-3))). - Bob Selcoe, Mar 17 2014
a(n) = sqrt(5)^(n-1)*(3*S(n-1, sqrt(5)) - sqrt(5)*S(n-2, sqrt(5))) with Chebyshev's S-polynomials (see A049310), where S(-1, x) = 0 and S(-2, x) = -1. This is the (1,1) entry of A^n with the matrix A=(3,1;1,2). See the comment by David Neil McGrath, Nov 18 2014. - Wolfdieter Lang, Dec 04 2014
Conjecture: a(n) = 2*a(n-1) + A039717(n). - Benito van der Zander, Nov 20 2015
a(n) = A189315(n+1) / 10. - Tom Copeland, Dec 08 2015
a(n) = A093129(n) + A030191(n-1). - Gary W. Adamson, Apr 24 2023
E.g.f.: exp(5*x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Jun 03 2024

A026378 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=1; also a(n) = T(2n-1,n-1).

Original entry on oeis.org

1, 4, 17, 75, 339, 1558, 7247, 34016, 160795, 764388, 3650571, 17501619, 84179877, 406020930, 1963073865, 9511333155, 46169418195, 224484046660, 1093097083475, 5329784874185, 26018549129545, 127154354598330, 622031993807565

Offset: 1

Clark Kimberling

nonn

Number of lattice paths from (0,0) to the line x=n-1 that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and three types of steps H=(1,0) (left factors of 3-Motzkin steps). Example: a(3)=17 because we have UD, UU, 9 HH paths, 3 HU paths and 3 UH paths. - Emeric Deutsch, Jan 22 2004
Also a(n) = number of integer strings s(0), ..., s(n) counted by array U in A026386 that have s(n)=1; a(n) = U(2n-1, n-1).
The Hankel transform of [1,1,4,17,75,339,1558,...] is [1,3,8,21,55,144,377,...] (see A001906). - Philippe Deléham, Apr 13 2007
Number of peaks in all skew Dyck paths of semilength n. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=4 because in the 3 (=A002212(2)) skew Dyck paths (UD)(UD), U(UD)D and U(UD)L we have altogether 4 peaks (shown between parentheses). - Emeric Deutsch, Jul 25 2007
Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007
5th binomial transform of (-1)^n*A000108. - Paul Barry, Jan 13 2009
From Gary W. Adamson, May 17 2009: (Start)
Convolved with A007317, (1, 2, 5, 15, 51, ...) = A026376: (1, 6, 30, 144, ...)
Equals A026375, (1, 3, 11, 45, 195, ...) convolved with A002212 prefaced with
a 1: (1, 1, 3, 10, 36, 137, ...). (End)
From Tom Copeland, Nov 09 2014: (Start)
The array belongs to an interpolated family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the interpolating o.g.f. [1-sqrt(1-4x/(1+(1-t)x))]/2 and inverse x(1-x)/[1+(t-1)x(1-x)]. See A091867 for more info on this family. Here the interpolation is t=-4 (mod signs in the results).
Let C(x) = [1 - sqrt(1-4x)]/2, an o.g.f. for the Catalan numbers A000108, with inverse Cinv(x) = x*(1-x) and P(x,t) = x/(1+t*x) with inverse P(x,-t).
O.g.f: G(x) = [-1 + sqrt(1 + 4*x/(1-5x))]/2 = -C[P(-x,5)].
Inverse O.g.f: Ginv(x) = x*(1+x)/[1 + 5x*(1+x)] = -P(Cinv(-x),-5) (signed A039717). (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Shu-Chiuan Chang, Robert Shrock, Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips, J. Stat. Physics 137 (2009) 667, table 5.
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Toufik Mansour, Jose Luis Ramirez, Enumration of Fuss-skew paths, Ann. Math. Inform. 55 (2022) 125-136, table 2, l=1.
László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.

Half the values of A026387. Bisection of A026380 and A026392.
Cf. A026375, A026376, A007317, A002212. - Gary W. Adamson, May 17 2009
Cf. A000108, A005043, A026375, A026380, A039717, A091867, A126182.

a := n -> (-1)^n*simplify(GegenbauerC(n-2,-n+1,3/2) - GegenbauerC(n-1,-n+1,3/2)): seq(a(n), n=1..23); # Peter Luschny, May 13 2016

CoefficientList[Series[(1/2)/(5*x^2-x)*(1-5*x-(1-6*x+5*x^2)^(1/2)),{x,0,30}],x] (* Vincenzo Librandi, May 13 2012 *)
Table[Hypergeometric2F1[3/2, 1-n, 2, -4], {n, 1, 20}] (* Vladimir Reshetnikov, Apr 25 2016 *)

G.f.: (1/2)/(5*x^2-x)*(1-5*x-(1-6*x+5*x^2)^(1/2)). E.g.f.: exp(3*x)*(BesselI(0, 2*x)+BesselI(1, 2*x)). - Vladeta Jovovic, Oct 03 2003
G.f.: [(1-z)/sqrt(1-6z+5z^2)-1]/2 = z + 4z^2 + 17z^3 + ... - Emeric Deutsch, Jan 22 2004
a(n) = coefficient of t^n in (1+t)(1+3t+t^2)^(n-1). - Emeric Deutsch, Jan 30 2004
a(n) = A026380(2n-2). - Emeric Deutsch, Feb 18 2004
a(n) = [2(3n-2)a(n-1) - 5(n-2)a(n-2)]/n for n>=2; a(0)=0, a(1)=1. - Emeric Deutsch, Mar 18 2004
a(n+1) = sum(k=0, n, binomial(n, k)*sum(i=0, k, binomial(k+i, i))). - Benoit Cloitre, Aug 06 2004
a(n+1) = sum(k=0, n, binomial(n, k)*binomial(2*k+1, k+1)). - Benoit Cloitre, Aug 06 2004
a(n) = Sum(k*A126182(n-1,k-1),k=1..n). - Emeric Deutsch, Jul 25 2007
From Paul Barry, Jan 13 2009: (Start)
G.f.: (1/(1-5x))*c(-x/(1-5x)), c(x) the g.f. of A000108;
a(n) = sum{k=0..n, C(n,k)*(-1)^k*A000108(k)*5^(n-k)} (offset 0). (End)
G.f. 1/(1 - 3x - x(1 - x)/(1 - x - x(1 - x)/(1 - x - x(1 - x)/(1 - x - x(1 - x)/(1...(continued fraction). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Jul 02 2010
a(n) ~ 5^(n-1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 08 2012
a(n) = hypergeom([3/2, 1-n], [2], -4). - Vladimir Reshetnikov, Apr 25 2016
a(n) = (-1)^n*(GegenbauerC(n-2,-n+1,3/2) - GegenbauerC(n-1,-n+1,3/2)). - Peter Luschny, May 13 2016

A226205 a(n) = F(n)^2 - F(n-1)^2 or F(n+1) * F(n-2) where F(n) = A000045(n), the Fibonacci numbers.

Original entry on oeis.org

1, 0, 3, 5, 16, 39, 105, 272, 715, 1869, 4896, 12815, 33553, 87840, 229971, 602069, 1576240, 4126647, 10803705, 28284464, 74049691, 193864605, 507544128, 1328767775, 3478759201, 9107509824, 23843770275, 62423800997, 163427632720, 427859097159, 1120149658761

Offset: 1

Michael Somos, Jun 06 2013

A001519(n)^2 = A079472(n)^2 + a(n)^2 and (A001519(n), A079472(n), a(n)) is a Pythagorean triple.
INVERT transform is A052156. PSUM transform is A007598. SUMADJ transform is A088305. BINOMIAL transform is A039717. BINOMIAL transform with 0 prepended is A112091 with 0 prepended. BINOMIAL transform inverse is A084179(n+1).
In general, the difference between squares of two consecutive terms of a second order linear recurrence having a signature of (c,d) will be a third order recurrence with signature (c^2+d,(c^2+d)*d,-d^3). - Gary Detlefs, Mar 13 2025

			G.f. = x + 3*x^3 + 5*x^4 + 16*x^5 + 39*x^6 + 105*x^7 + 272*x^8 + 715*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
John P. Bonomo and Montana Ferita, A Small Fib, College Math. J., 2023.
Nurettin Irmak, Product of arbitrary Fibonacci numbers with distance 1 to Fibonomial coefficient, Turk J Math, (2017) 41: 825-828. See p. 828.
C.-A. Laisant, Observations sur les triangles rectangles en nombres entiers et les suites de Fibonacci, Nouvelles Annales de Math. (1919, in French) Série 4, Vol. 19, 391-397.
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A001519, A001622, A001654, A007598, A033999, A039717, A052156, A079472, A084179, A088305, A112091, A121646, A290565.
Cf. similar sequences of the type k*F(n)*F(n+1)+(-1)^n listed in A264080.
Cf. A260259: numbers of the form F(n)*F(n+1)-(-1)^n. - Bruno Berselli, Nov 02 2015

[Fibonacci(n)^2-Fibonacci(n-1)^2: n in [1..40]]; // Vincenzo Librandi, Jun 18 2014

a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1,0,3>>)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 28 2016

a[ n_] := Fibonacci[n + 1] Fibonacci[n - 2]; (* Michael Somos, Jun 17 2014 *)
CoefficientList[Series[(1 - x)^2/((1 + x) (1 - 3 x + x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2014 *)

{a(n) = fibonacci( n + 1) * fibonacci( n - 2)};

a(n) = round(2^(-1-n)*(-(-1)^n*2^(3+n)-(3-sqrt(5))^n*(1+sqrt(5))+(-1+sqrt(5))*(3+sqrt(5))^n)/5) \\ Colin Barker, Sep 28 2016

lista(nn) = {my(p = (3*x-1)/(x^3-2*x^2-2*x+1)); for (n=1, nn, p = deriv(p, x); print1(subst(p, x, 0)/n!, ", "); ); } \\ Michel Marcus, May 22 2018

G.f.: x * (1 - x)^2 / ((1 + x) * (1 -3*x + x^2)).
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n) = - A121646(n).
a(n) = -a(1-n) for all n in Z.
a(n) = A121801(n+1) / 2. - Michael Somos, Jun 17 2014
a(n) = a(n-1) + A000045(n-1)^2 - 2*(-1)^n, for n>1. - Alexander Samokrutov, Sep 07 2015
a(n) = F(n-1)*F(n) - (-1)^n. - Bruno Berselli, Oct 30 2015
a(n) = 2^(-1-n)*(-(-1)^n*2^(3+n)-(3-sqrt(5))^n*(1+sqrt(5))+(-1+sqrt(5))*(3+sqrt(5))^n)/5. - Colin Barker, Sep 28 2016
From Amiram Eldar, Oct 06 2020: (Start)
Sum_{n>=3} 1/a(n) = (1/2) * A290565 - 1/4.
Sum_{n>=3} (-1)^(n+1)/a(n) = (3/2) * (1/phi - 1/2), where phi is the golden ratio (A001622). (End)

A030523 A convolution triangle of numbers obtained from A001792.

Original entry on oeis.org

1, 3, 1, 8, 6, 1, 20, 25, 9, 1, 48, 88, 51, 12, 1, 112, 280, 231, 86, 15, 1, 256, 832, 912, 476, 130, 18, 1, 576, 2352, 3276, 2241, 850, 183, 21, 1, 1280, 6400, 10976, 9424, 4645, 1380, 245, 24, 1, 2816, 16896, 34848, 36432, 22363, 8583, 2093, 316, 27, 1

Offset: 1

Wolfdieter Lang

a(n,m) := s1p(3; n,m), a member of a sequence of unsigned triangles including s1p(2; n,m)= A007318(n-1,m-1) (Pascal's triangle). Signed version: (-1)^(n-m)*a(n,m) := s1(3; n,m).
With offset 0, this is T(n,k) = Sum_{i=0..n} C(n,i)*C(i+k+1,2k+1). Binomial transform of A078812 (product of lower triangular matrices). - Paul Barry, Jun 22 2004
Subtriangle of the triangle T(n,k) given by (0, 3, -1/3, 4/3, 0, 0, 0, 0, 0, 0, 0, ... ) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 20 2013

			{1}; {3,1}; {8,6,1}; {20,25,9,1}; {48,88,51,12,1}; ...
(0, 3, -1/3, 4/3, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
1
0   1
0   3   1
0   8   6   1
0  20  25   9   1
0  48  88  51  12   1
...
- _Philippe Deléham_, Feb 20 2013

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First ten rows.

Cf. A057682 (alternating row sums).

a[n_, m_] := SeriesCoefficient[(1-2*x)^2/((x^2-x)*y + (1-2*x)^2) - 1, {x, 0, n}, {y, 0, m}]; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Apr 28 2015, after Vladimir Kruchinin *)

a(n, 1) = A001792(n-1).
Row sums = A039717(n).
a(n, m) = 2*(2*m+n-1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, nT(n,k) = 4*T(n-1,k) - 4*T(n-2,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Feb 20 2013
Sum_{k=1..n} T(n,k)*2^(k-1) = A140766(n). -Philippe Deléham, Feb 20 2013
G.f.: (1-2*x)^2/((x^2-x)*y+(1-2*x)^2)-1. - Vladimir Kruchinin, Apr 28 2015

A216219 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=5 or if k-n>=5, T(4,0) = T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 0, 5, 10, 10, 5, 0, 0, 5, 15, 20, 15, 5, 0, 0, 0, 20, 35, 35, 20, 0, 0, 0, 0, 20, 55, 70, 55, 20, 0, 0, 0, 0, 0, 75, 125, 125, 75, 0, 0, 0, 0, 0, 0, 75, 200, 250, 200

Offset: 0

Philippe Deléham, Mar 13 2013

			Square array begins:
1, 1,  1,  1,   1,   0,   0,   0,    0,    0, 0, ...
1, 2,  3,  4,   5,   5,   0,   0,    0,    0, 0, ...
1, 3,  6, 10,  15,  20,  20,   0,    0,    0, 0, ...
1, 4, 10, 20,  35,  55,  75,  75,    0,    0, 0, ...
1, 5, 15, 35,  70, 125, 200, 275,  275,    0, 0, ...
0, 5, 20, 55, 125, 250, 450, 725, 1000, 1000, 0, ...
0, 0, 20, 75, 200, 450, 900, ...

Cf. A030191, A039717, A068913, A081567, A147748, A216210, A216212, A216218

T(n,n) = A147748(n).
T(n+1,n) = T(n,n+1) = A081567(n).
T(n+2,n) = T(n,n+2) = A039717(n+1).
T(n+3,n) = T(n+4,n) = T(n,n+3) = T(n,n+4) = A030191(n).
Sum_{k, 0<=k<=n} T(n-k,k) = A068913(4,n) = A216212(n).

A216212 Number of n step walks (each step +-1 starting from 0) which are never more than 4 or less than -4.

Original entry on oeis.org

1, 2, 4, 8, 16, 30, 60, 110, 220, 400, 800, 1450, 2900, 5250, 10500, 19000, 38000, 68750, 137500, 248750, 497500, 900000, 1800000, 3256250, 6512500, 11781250, 23562500, 42625000, 85250000, 154218750, 308437500, 557968750, 1115937500, 2018750000, 4037500000

Offset: 0

Philippe Deléham, Mar 13 2013

The number of n step walks (each step +-1 starting from 0) which are never more than k or less than -k is given by a(n,k) = 2^n/(k+1)*Sum_{r=1..k+1} (-1)^r*cos((Pi*(2*r-1))/(2*(k+1)))^n*cot((Pi*(1-2*r))/(4*(k+1))), n<>0 if k even. Here we have k=4. - Herbert Kociemba, Sep 22 2020

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-5).

Cf. A068911, A068912, A068913, A178381 (starting from 4).

nn=30;CoefficientList[Series[(1+x-x^2)^2/(1-5x^2+5x^4),{x,0,nn}],x] (* Geoffrey Critzer, Jan 14 2014 *)
a[0,4]=1; a[n_,k_]:=2^n/(k+1) Sum[(-1)^r Cos[(Pi (2r-1))/(2 (k+1))]^n Cot[(Pi (1-2r))/(4 (k+1))],{r,1,k+1}]
Table[a[n,4],{n,0,40}]//Round (* Herbert Kociemba, Sep 22 2020 *)

a(n) = A068913(4,n).
G.f.: (1+2*x-x^2-2*x^3+x^4)/(1-5*x^2+5*x^4).
a(n) = 5*a(n-2) - 5*a(n-4), a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8, a(4) = 16.
a(2*n+1) = 2*A039717(n+1), a(2*n+2) = 4*A039717(n+1).
a(n) = (2^n/5)*Sum_{r=1..5} (-1)^r*cos(Pi*(2*r-1)/10)^n*cot(Pi*(1-2*r)/20), n>0. - Herbert Kociemba, Sep 22 2020

A109106 a(n) = (1/sqrt(5))((sqrt(5) + 1)((15 + 5sqrt(5))/2)^(n-1) + (sqrt(5) - 1)((15 - 5*sqrt(5))/2)^(n-1)).

Original entry on oeis.org

2, 20, 250, 3250, 42500, 556250, 7281250, 95312500, 1247656250, 16332031250, 213789062500, 2798535156250, 36633300781250, 479536132812500, 6277209472656250, 82169738769531250, 1075615844726562500

Offset: 1

Emeric Deutsch, Jun 19 2005

nonn

Kekulé numbers for certain benzenoids.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 215, K{T_m}).

Cf. A179135. - Johannes W. Meijer, Jul 01 2010

a:=n->(1/sqrt(5))*((sqrt(5)+1)*((15+5*sqrt(5))/2)^(n-1)+(sqrt(5)-1)*((15-5*sqrt(5))/2)^(n-1)): seq(expand(a(n)),n=1..19);

G.f.: 2z(1-5z)/(1 - 15z + 25z^2).
From Johannes W. Meijer, Jul 01 2010: (Start)
a(n) = A178381(4*n+2).
Lim_{k->infinity} a(n+k)/a(k) = (A020876(2*n) + 5*A039717(2*n-2)*sqrt(5))/2.
Lim_{n->infinity} A020876(2*n)/(5*A039717(2*n-2)) = sqrt(5).
(End)
a(n) = 2*5^(n-1)*Fibonacci(2*n-1). - Ehren Metcalfe, Apr 21 2018

A217777 Expansion of (1+x)(1+2x)(1-x)/(1-5x^2+5*x^4).

Original entry on oeis.org

1, 2, 4, 8, 15, 30, 55, 110, 200, 400, 725, 1450, 2625, 5250, 9500, 19000, 34375, 68750, 124375, 248750, 450000, 900000, 1628125, 3256250, 5890625, 11781250, 21312500, 42625000, 77109375, 154218750, 278984375, 557968750, 1009375000, 2018750000, 3651953125

Offset: 0

Philippe Deléham, Mar 24 2013

Index entries for linear recurrences with constant coefficients, signature (0, 5, 0, -5).

Cf. A217770

Vec((1+x)*(1+2*x)*(1-x)/(1-5*x^2+5*x^4)+O(x^66)) /* Joerg Arndt, Mar 29 2013 */

a(n) = A216212(n+1)/2.
a(2n) = A039717(n+1), a(2n+1) = 2*a(2n) = 2*A039717(n+1).
a(n) = sum(A217770(n-k,k), 0<=k<=n).
a(n) = 5*a(n-2) - 5*a(n-4) for n>=4, a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8.
G.f.: (1+x)*(1+2*x)*(1-x)/(1-5*x^2+5*x^4).

Corrected name (g.f.), Joerg Arndt, Mar 29 2013

Showing 1-10 of 12 results. Next

cp's OEIS Frontend

A030528 Triangle read by rows: a(n,k) = binomial(k,n-k).

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A030191 Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2.

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A081567 Second binomial transform of F(n+1).

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A026378 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=1; also a(n) = T(2n-1,n-1).

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A226205 a(n) = F(n)^2 - F(n-1)^2 or F(n+1) * F(n-2) where F(n) = A000045(n), the Fibonacci numbers.

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A030523 A convolution triangle of numbers obtained from A001792.

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A109106 a(n) = (1/sqrt(5))((sqrt(5) + 1)((15 + 5sqrt(5))/2)^(n-1) + (sqrt(5) - 1)((15 - 5*sqrt(5))/2)^(n-1)).

A217777 Expansion of (1+x)(1+2x)(1-x)/(1-5x^2+5*x^4).