A026374 -id:A026374 - cp's OEIS frontend

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).

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

A026376 a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=2; also a(n) = T(2n,n-1).

Original entry on oeis.org

1, 6, 30, 144, 685, 3258, 15533, 74280, 356283, 1713690, 8263596, 39938616, 193419915, 938430990, 4560542550, 22195961280, 108171753355, 527816696850, 2578310320610, 12607504827600, 61706212037295, 302275142049870, 1481908332595625, 7270432009471224

Offset: 1

Clark Kimberling

Number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis) from (0,0) to (2n+2,0), with exactly one peak at an even level. E.g., a(2)=6 because we have UUDDH, HUUDD, UDUUDD, UUDDUD, UUDHD and UHUDD. - Emeric Deutsch, Dec 28 2003
Number of left steps in all skew Dyck paths of semilength n+1. 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)=6 because in the 10 (=A002212(3)) skew Dyck paths of semilength 3 ( namely UDUUDL, UUUDLD, UUDUDL, UUUDDL, UUUDLL and five Dyck paths that have no left steps) we have altogether 6 left steps. - Emeric Deutsch, Aug 05 2007
From Gary W. Adamson, May 17 2009: (Start)
Equals A026378 (1, 4, 17, 75, ...) convolved with A007317 (1, 2, 5, 15, 51, ...).
Equals A081671 (1, 3, 11, 45, ...) convolved with A002212 (1, 3, 10, 36, 137, ...).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 18.
Toufik Mansour and José Luis Ramírez, Enumeration of Fuss-skew paths, Ann. Math. Inform. (2022) Vol. 55, 125-136. See p. 129.

Cf. A006318, A002212, A081671, A007317, A026378.

a := n -> simplify(GegenbauerC(n-1, -n, -3/2)):
seq(a(n), n=1..24); # Peter Luschny, May 09 2016

Rest[CoefficientList[Series[(1-3*x-Sqrt[1-6*x+5*x^2])/(2*x*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 13 2014 *)

a(n)=if(n<0,0,polcoeff((1+3*x+x^2)^n,n-1))

A026376 = lambda n : n*hypergeometric([1, 3/2, 1-n], [1, 3], -4)
[round(A026376(n).n(100)) for n in (1..24)] # Peter Luschny, Sep 16 2014

# Recurrence:
def A026376():
    x, y, n = 1, 1, 1
    while True:
        x, y = y, ((6*n + 3)*y - (5*n - 5)*x) / (n + 2)
        yield n*x
        n += 1
a = A026376()
[next(a) for i in (1..24)] # Peter Luschny, Sep 16 2014

E.g.f.: exp(3x)*I_1(2x), where I_1 is Bessel function. - Michael Somos, Sep 09 2002
G.f.: (1 - 3*z - t)/(2*z*t) where t = sqrt(1-6*z+5*z^2). - Emeric Deutsch, May 25 2003
a(n) = [t^(n+1)](1+3t+t^2)^n. a := n -> Sum_{j=ceiling((n+1)/2)..n} 3^(2j-n-1)*binomial(n, j)*binomial(j, n+1-j). - Emeric Deutsch, Jan 30 2004
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2k, k+1). - Paul Barry, Sep 20 2004
a(n) = n*A002212(n). - Emeric Deutsch, Aug 05 2007
D-finite with recurrence (n+1)*a(n) - 9*n*a(n-1) + (23*n-27)*a(n-2) + 15*(-n+2)*a(n-3) = 0. - R. J. Mathar, Dec 02 2012
a(n) ~ 5^(n+1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 13 2014
a(n) = n*hypergeometric([1, 3/2, 1-n],[1, 3],-4). - Peter Luschny, Sep 16 2014
a(n) = GegenbauerC(n-1, -n, -3/2). - Peter Luschny, May 09 2016

A026377 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=4; also a(n) = T(2n,n-2).

Original entry on oeis.org

1, 9, 58, 330, 1770, 9198, 46928, 236736, 1185645, 5909805, 29362806, 145570230, 720606705, 3563543025, 17610412600, 86989143480, 429579843435, 2121099312195, 10472653252550, 51708363376950, 255326054688320, 1260886172311524, 6227515552731528, 30762417293645400

Offset: 2

Clark Kimberling

Number of lattice paths from (0,0) to (2n,4), using steps U=(1,1), D=(1,-1) and at even levels(zero, positive and negative) also H=(2,0). Example: a(3)=9 because we have UUUUUD, UUUUDU, UUUDUU, UUDUUU, UDUUUU, DUUUUU, HUUUU, UUHUU and UUUUH. - Emeric Deutsch, Jan 30 2004

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A026374.

series( (3*x-1+(1-6*x+7*x^2)/sqrt(5*x^2-6*x+1))/(2*x^2), x=0, 30); # Mark van Hoeij, Apr 18 2013

CoefficientList[Series[(3*x-1+(1-6*x+7*x^2)/Sqrt[5*x^2-6*x+1])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 07 2013 *)

x='x+O('x^66); Vec((3*x-1+(1-6*x+7*x^2)/sqrt(5*x^2-6*x+1))/(2*x^2)) /* Joerg Arndt, Apr 19 2013 */

From Emeric Deutsch, Jan 30 2004: (Start)
a(n) = [t^(n+2)](1+3t+t^2)^n.
a(n) = Sum_{j=ceiling((n+2)/2)..n} (3^(2j-n-2)*binomial(n, j)*binomial(j, n+2-j)). (End)
From Paul Barry, Sep 20 2004: (Start)
E.g.f.: exp(3x) * BesselI(2, 2x).
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2k, k+2). (End)
Conjecture: n*(n+4)*a(n) - 3*(n+2)*(2*n+3)*a(n-1) + 5*(n+2)*(n+1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
G.f.: (3*x - 1 + (1-6*x+7*x^2)/sqrt(5*x^2-6*x+1))/(2*x^2). - Mark van Hoeij, Apr 18 2013
a(n) ~ 5^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 07 2013
Assuming offset 0: a(n) = C(2*n+4,n)*hypergeom([-n,-n-4],[-3/2-n],-1/4). - Peter Luschny, May 09 2016

A026385 Sum{T(n-k,k)}, 0<=k<=[ n/2 ], where T is the array in A026374.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 20, 33, 59, 104, 178, 314, 549, 952, 1669, 2913, 5074, 8872, 15482, 27007, 47172, 82325, 143675, 250848, 437822, 764198, 1334041, 2328512, 4064457, 7094833, 12384034, 21616716, 37732990, 65863651

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,3,1).

I:=[1,1,2,4]; [n le 4 select I[n] else Self(n-2)+3*Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 19 2014

LinearRecurrence[{0,1,3,1},{1,1,2,4},40] (* Harvey P. Dale, Jun 18 2014 *)
CoefficientList[Series[(1 + x + x^2)/(1 - x^2 - 3 x^3 - x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2014 *)

G.f.: (1+x+x^2)/(1-x^2-3x^3-x^4). - Ralf Stephan, Apr 30 2004

a(n) = a(n-2) + 3*a(n-3) + a(n-4) for n>3. - Vincenzo Librandi, Jun 19 2014

A026379 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=3; also a(n) = T(2n-1,n-2).

Original entry on oeis.org

1, 7, 39, 202, 1015, 5028, 24731, 121208, 593019, 2899335, 14173401, 69301422, 338990145, 1659037695, 8124085575, 39806373880, 195160896835, 957396540285, 4699409632805, 23080158080150, 113414575414245, 557601196738190

Offset: 2

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 2..300

Partial sums of A034942.

sum(binomial(n,k)*binomial(2*k+1,k-1),k=0..n); n=0,1,... # N. J. A. Sloane

a[n_] := (n-1)*Hypergeometric2F1[5/2, 2-n, 4, -4]; Table[a[n], {n, 2, 23}](* Jean-François Alcover, Jun 12 2012, after N. J. A. Sloane *)
Table[Sum[Binomial[n,k]Binomial[2k+1,k-1],{k,0,n}],{n,30}] (* Harvey P. Dale, Apr 28 2013  *)

a(n) = [t^(n+1)]{(1+t)(1+3t+t^2)^(n-1)}. - Emeric Deutsch, Jan 30 2004

G.f.: -1/x+2-2*x^2/(10*x^2+sqrt(1-5*x)*sqrt(1-x)*(4*x-1)-7*x+1). - Vladimir Kruchinin, Aug 11 2015

A026380 a(n) = T(n,[ n/2 ]), where T is the array in A026374.

Original entry on oeis.org

1, 3, 4, 11, 17, 45, 75, 195, 339, 873, 1558, 3989, 7247, 18483, 34016, 86515, 160795, 408105, 764388, 1936881, 3650571, 9238023, 17501619, 44241261, 84179877, 212601015, 406020930, 1024642875, 1963073865, 4950790605

Offset: 0

Clark Kimberling

nonn

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

Cf. A026375, A026378.

a(2n)=A026378(n+1), a(2n-1)=A026375(n). - Emeric Deutsch, Feb 18 2004
a(2n) = A026378(2n+1), a(2n+1) = A026375(n+1).
Davenport et al. give a g.f.

A026381 T(n,n-2), where T is the array in A026374.

Original entry on oeis.org

1, 4, 11, 17, 30, 39, 58, 70, 95, 110, 141, 159, 196, 217, 260, 284, 333, 360, 415, 445, 506, 539, 606, 642, 715, 754, 833, 875, 960, 1005, 1096, 1144, 1241, 1292, 1395, 1449, 1558, 1615, 1730, 1790, 1911, 1974, 2101, 2167, 2300

Offset: 2

Clark Kimberling

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

a026381 = flip a026374 2  -- Reinhard Zumkeller, Feb 22 2014

Drop[CoefficientList[Series[z^2(1+3z+5z^2)/((1-z)^3(1+z)^2),{z,0,50}],z],2] (* or *) LinearRecurrence[{1,2,-2,-1,1},{1,4,11,17,30},50] (* Harvey P. Dale, Dec 31 2021 *)

G.f.: z^2*(1+3z+5z^2)/[(1-z)^3*(1+z)^2]. - Emeric Deutsch, Jan 25 2004

A026382 a(n) = T(n,n-3), where T is the array in A026374.

Original entry on oeis.org

1, 6, 17, 45, 75, 144, 202, 330, 425, 630, 771, 1071, 1267, 1680, 1940, 2484, 2817, 3510, 3925, 4785, 5291, 6336, 6942, 8190, 8905, 10374, 11207, 12915, 13875, 15840, 16936, 19176, 20417, 22950, 24345, 27189, 28747, 31920

Offset: 3

Clark Kimberling

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 3..1000

Cf. A026374.

a026382 = flip a026374 3  -- Reinhard Zumkeller, Feb 22 2014

G.f.: z^3*(1+5z+8z^2+13z^3)/[(1-z)^4*(1+z)^3]. - Emeric Deutsch, Jan 25 2004

A026384 a(n) = Sum_{j=0..i, i=0..n} T(i,j), where T is the array in A026374.

Original entry on oeis.org

1, 3, 8, 18, 43, 93, 218, 468, 1093, 2343, 5468, 11718, 27343, 58593, 136718, 292968, 683593, 1464843, 3417968, 7324218, 17089843, 36621093, 85449218, 183105468, 427246093, 915527343, 2136230468, 4577636718, 10681152343, 22888183593, 53405761718

Offset: 0

Clark Kimberling

Partial sums of A026383. Number of lattice paths from (0,0) that do not go to right of the line x=n, using the steps U=(1,1), D=(1,-1) and, at levels ...,-4,-2,0,2,4,..., also H=(2,0). Example: a(2)=8 because we have the empty path, U, D, UU, UD, DD, DU and H. - Emeric Deutsch, Feb 18 2004

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

Cf. A026383.

I:=[1,3,8]; [n le 3 select I[n] else Self(n-1)+5*Self(n-2)-5*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Aug 09 2017

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=5*a[n-2]+3 od: seq(a[n], n=1..29); # Zerinvary Lajos, Mar 17 2008

CoefficientList[Series[(1 + 2 x) / ((1 - x) (1 - 5 x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 09 2017 *)
LinearRecurrence[{1,5,-5},{1,3,8},40] (* Harvey P. Dale, May 31 2023 *)

Vec((2*x + 1)/(5*x^3 - 5*x^2 - x + 1) + O(x^40)) \\ Colin Barker, Nov 25 2016

G.f.: (1+2*x) / ((1-x)*(1-5*x^2)). - Ralf Stephan, Apr 30 2004
From Colin Barker, Nov 25 2016: (Start)
a(n) = (7*5^(n/2) - 3)/4 for n even.
a(n) = 3*(5^((n+1)/2) - 1)/4 for n odd.
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) for n>2.
(End)

A026950 a(n) = Sum_{k=0..n} (k+1) * T(n,k), with T given by A026374.

Original entry on oeis.org

1, 3, 10, 25, 75, 175, 500, 1125, 3125, 6875, 18750, 40625, 109375, 234375, 625000, 1328125, 3515625, 7421875, 19531250, 41015625, 107421875, 224609375, 585937500, 1220703125, 3173828125, 6591796875, 17089843750, 35400390625, 91552734375, 189208984375, 488281250000

Offset: 0

Clark Kimberling

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-25).

Cf. A026374.

a(n) = (n + 2) * (7 + 3*(-1)^n) * 5^((n-1)\2) / 4 \\ Andrew Howroyd, Dec 27 2024

a(n) = (n + 2) * (7 + 3*(-1)^n) * 5^floor((n-1)/2) / 4.
From Colin Barker, Oct 13 2012: (Start)
a(n) = 10*a(n-2) - 25*a(n-4).
G.f.: -(5*x^3-3*x-1)/(5*x^2-1)^2. (End)

a(28) onwards from Andrew Howroyd, Dec 27 2024

cp's OEIS Frontend

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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A026376 a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=2; also a(n) = T(2n,n-1).

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A026377 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=4; also a(n) = T(2n,n-2).

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A026385 Sum{T(n-k,k)}, 0<=k<=[ n/2 ], where T is the array in A026374.

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Formula

A026379 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=3; also a(n) = T(2n-1,n-2).

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Maple

Mathematica

Formula

A026380 a(n) = T(n,[ n/2 ]), where T is the array in A026374.

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Formula

A026381 T(n,n-2), where T is the array in A026374.

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Haskell

Mathematica

Formula

A026382 a(n) = T(n,n-3), where T is the array in A026374.

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