A026386 -id:A026386 - cp's OEIS frontend

A026392 T(n,[ n/2 ]), where T is the array in A026386.

1, 2, 4, 8, 17, 34, 75, 150, 339, 678, 1558, 3116, 7247, 14494, 34016, 68032, 160795, 321590, 764388, 1528776, 3650571, 7301142, 17501619, 35003238, 84179877, 168359754, 406020930, 812041860, 1963073865, 3926147730

Offset: 1

Clark Kimberling

nonn

Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 15.

Cf. A026378, A085362.

A026392 := proc(n)
    A026386(n,floor(n/2)) ;
end proc: # R. J. Mathar, Feb 10 2015

Conjecture: (n+1)*a(n) +2*(n-1)*a(n-1) +2*(-3*n+1)*a(n-2) +4*(-3*n+7)*a(n-3) +5*(n-3)*a(n-4) +10*(n-5)*a(n-5)=0. - R. J. Mathar, Feb 10 2015

Offset corrected. R. J. Mathar, Feb 10 2015

A026387 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=0; also a(n) = T(2n,n).

Original entry on oeis.org

2, 8, 34, 150, 678, 3116, 14494, 68032, 321590, 1528776, 7301142, 35003238, 168359754, 812041860, 3926147730, 19022666310, 92338836390, 448968093320, 2186194166950, 10659569748370, 52037098259090, 254308709196660

Offset: 0

Clark Kimberling

László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.

Essentially the same as A085362.

Cf. A026378.

a(n) = A085362(n+1), n >= 0. - Hartmut F. W. Hoft, Jul 07 2024

A026388 a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=2; also a(n) = T(2n,n-1).

Original entry on oeis.org

1, 5, 24, 114, 541, 2573, 12275, 58747, 282003, 1357407, 6549906, 31675020, 153481299, 745011075, 3622111560, 17635418730, 85975792075, 419644943495, 2050493623760, 10029194506990, 49098707209695, 240568930012575

Offset: 1

Clark Kimberling

nonn

Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562. Table 1, y_n.
Toufik Mansour and José Luis Ramírez, Enumeration of Fuss-skew paths, Ann. Math. Inform. (2022) Vol. 55, 125-136. See p. 129, eq (2.1) at l=1.
László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.

Table[HypergeometricPFQ[{3/2, 2, 1-n}, {1, 3}, -4], {n, 1, 20}] (* Vladimir Reshetnikov, Apr 25 2016 *)

a(n) = hypergeom([3/2, 2, 1-n], [1, 3], -4). - Vladimir Reshetnikov, Apr 25 2016
D-finite with recurrence -(n+1)*(2*n-1)*a(n) +(12*n^2-2*n+1)*a(n-1) -5*(2*n+1)*(n-2)*a(n-2)=0. - R. J. Mathar, Jun 21 2018
G.f.: ((x-1)*sqrt((5*x-1)/(x-1))-3*x+1)/(2*x*sqrt((5*x-1)/(x-1))). - Vladimir Kruchinin, Sep 17 2020
a(n) = Sum_{k=1..n} C(2*k,k-1)*C(n-1,k-1). - Vladimir Kruchinin, Sep 17 2020
a(n) ~ 2 * 5^(n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Sep 17 2020

A026389 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=4; also a(n) = T(2n,n-2).

Original entry on oeis.org

1, 8, 49, 272, 1440, 7428, 37730, 189808, 948909, 4724160, 23453001, 116207424, 575036475, 2842936320, 14046869575, 69378730880, 342590699955, 1691519468760, 8351553940355, 41235710124400, 203617691311370, 1005560117623204

Offset: 2

Clark Kimberling

nonn

Binomial transform of A002694. - Ross La Haye, Mar 05 2005

Conjecture: (n+2)*a(n) +4*(-3*n-2)*a(n-1) +2*(24*n-19)*a(n-2) +4*(-18*n+43)*a(n-3) +35*(n-4)*a(n-4)=0. - R. J. Mathar, May 29 2013

A026393 a(n) = T(n,n-2), where T is the array in A026386.

Original entry on oeis.org

1, 4, 8, 17, 24, 39, 49, 70, 83, 110, 126, 159, 178, 217, 239, 284, 309, 360, 388, 445, 476, 539, 573, 642, 679, 754, 794, 875, 918, 1005, 1051, 1144, 1193, 1292, 1344, 1449, 1504, 1615, 1673, 1790, 1851, 1974, 2038, 2167, 2234, 2369, 2439, 2580, 2653, 2800

Offset: 2

Clark Kimberling

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

Vec(x^2*(1+3*x+2*x^2+3*x^3)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 26 2016

G.f.: x^2*(1+3*x+2*x^2+3*x^3) / ((1-x)^3*(1+x)^2). - Emeric Deutsch, Feb 18 2004
From Colin Barker, Jan 26 2016: (Start)
a(n) = (18*n^2-46*n+(-1)^n*(13-6*n)+35)/16.
a(n) = (9*n^2-26*n+24)/8 for n even.
a(n) = (9*n^2-20*n+11)/8 for n odd.
(End)

A026394 a(n) = T(n,n-3), where T is the array in A026386.

Original entry on oeis.org

1, 5, 17, 34, 75, 114, 202, 272, 425, 535, 771, 930, 1267, 1484, 1940, 2224, 2817, 3177, 3925, 4370, 5291, 5830, 6942, 7584, 8905, 9659, 11207, 12082, 13875, 14880, 16936, 18080, 20417, 21709, 24345, 25794, 28747, 30362, 33650, 35440, 39081, 41055, 45067

Offset: 3

Clark Kimberling

Colin Barker, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1)

t[n_, 0] := 1; t[n_, n_] := 1; t[n_, k_] := t[n, k] = Which[EvenQ@ n, t[n - 1, k - 1] + t[n - 1, k], OddQ@ n, t[n - 1, k - 1] + t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, n - 3], {n, 3, 45}] (* Michael De Vlieger, Jan 29 2016, after Clark Kimberling at A026386 *)

Vec(x^3*(1+4*x+9*x^2+5*x^3+8*x^4)/((1-x)^4*(1+x)^3) + O(x^100)) \\ Colin Barker, Jan 29 2016

G.f.: x^3*(1+4*x+9*x^2+5*x^3+8*x^4) / ((1-x)^4*(1+x)^3). - Emeric Deutsch, Feb 18 2004
From Colin Barker, Jan 29 2016: (Start)
a(n) = (18*n^3-9*(-1)^n*n^2-111*n^2+53*(-1)^n*n+243*n-75*(-1)^n-181)/32.
a(n) = (9*n^3-60*n^2+148*n-128)/16 for n even.
a(n) = (9*n^3-51*n^2+95*n-53)/16 for n odd.
(End)

A026396 Sum_{T(i,j)}, 0<=j<=i, 0<=i<=n, where T is the array in A026386.

Original entry on oeis.org

3, 7, 17, 37, 87, 187, 437, 937, 2187, 4687, 10937, 23437, 54687, 117187, 273437, 585937, 1367187, 2929687, 6835937, 14648437, 34179687, 73242187, 170898437, 366210937, 854492187, 1831054687, 4272460937, 9155273437, 21362304687, 45776367187, 106811523437

Offset: 0

Clark Kimberling

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

Cf. A026386.

LinearRecurrence[{1, 5, -5}, {3, 7, 17}, 50] (* Paolo Xausa, Sep 16 2024 *)

Vec((-5*x^2 + 4*x + 3)/(5*x^3 - 5*x^2 - x + 1) + O(x^40)) \\ Colin Barker, Nov 25 2016

G.f.: (3+4*x-5*x^2) / ((1-x)*(1-5*x^2)). - Ralf Stephan, Apr 30 2004
From Colin Barker, Nov 25 2016: (Start)
a(n) = (7*5^(n/2) - 1)/2 for n even.
a(n) = (6*5^((n+1)/2) - 2)/4 for n odd.
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) for n>2. (End)
a(n) = (3-(-1)^n-(13+(-1)^n)*5^((1-(-1)^n+2*n)/4))/(2*(-1)^n-6). - Wesley Ivan Hurt, Oct 02 2021

A026955 a(n) = Sum_{k=0..n} (k+1) * T(n,k), with T given by A026386.

Original entry on oeis.org

1, 3, 8, 25, 60, 175, 400, 1125, 2500, 6875, 15000, 40625, 87500, 234375, 500000, 1328125, 2812500, 7421875, 15625000, 41015625, 85937500, 224609375, 468750000, 1220703125, 2539062500, 6591796875, 13671875000, 35400390625

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. A026386.

a(n) = if(n==0, 1, (n + 2) * (7 - 3*(-1)^n) * 5^floor(n/2) / 10) \\ Andrew Howroyd, Dec 27 2024

a(n) = (n + 2) * (7 - 3*(-1)^n) * 5^floor(n/2) / 10 for n > 0.
From Colin Barker, Oct 13 2012: (Start)
a(n) = 10*a(n-2) - 25*a(n-4) for n>4.
G.f.: (5*x^4-5*x^3-2*x^2+3*x+1)/(5*x^2-1)^2. (End)

A026397 Sum{T(n-k,k)}, 0<=k<=[ n/2 ], where T is the array in A026386.

Original entry on oeis.org

1, 2, 3, 6, 10, 17, 31, 53, 92, 163, 282, 492, 863, 1501, 2621, 4582, 7987, 13946, 24354, 42489, 74179, 129497, 226000, 394523, 688670, 1202020, 2098239, 3662553, 6392969, 11159290, 19478867, 34000750, 59349706, 103596641

Offset: 0

Clark Kimberling

nonn

Index entries for linear recurrences with constant coefficients, signature (0,1,3,1).

LinearRecurrence[{0,1,3,1},{1,2,3,6},40] (* Harvey P. Dale, Apr 15 2015 *)

Pairwise sums of A026385. - Ralf Stephan, Jul 22 2003

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

A026951 Self-convolution of array T given by A026386.

Original entry on oeis.org

1, 2, 6, 34, 116, 678, 2438, 14494, 53538, 321590, 1207186, 7301142, 27702096, 168359754, 643682106, 3926147730, 15096518580, 92338836390, 356629256930, 2186194166950, 8473375581420, 52037098259090, 202271610937570, 1244063987615130, 4847394356321770, 29851422385561898

Offset: 0

Clark Kimberling

nonn

More terms from Sean A. Irvine, Oct 17 2019

cp's OEIS Frontend

A026392 T(n,[ n/2 ]), where T is the array in A026386.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

Extensions

A026387 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=0; also a(n) = T(2n,n).

Views

Author

Keywords

Links

Crossrefs

Formula

A026388 a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=2; also a(n) = T(2n,n-1).

Views

Author

Keywords

Links

Programs

Mathematica

Formula

A026389 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=4; also a(n) = T(2n,n-2).

Views

Author

Keywords

Comments

Formula

A026393 a(n) = T(n,n-2), where T is the array in A026386.

Views

Author

Keywords

Links

Programs

PARI

Formula

A026394 a(n) = T(n,n-3), where T is the array in A026386.

Views

Author

Keywords

Links

Programs

Mathematica

PARI

Formula

A026396 Sum_{T(i,j)}, 0<=j<=i, 0<=i<=n, where T is the array in A026386.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A026955 a(n) = Sum_{k=0..n} (k+1) * T(n,k), with T given by A026386.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A026397 Sum{T(n-k,k)}, 0<=k<=[ n/2 ], where T is the array in A026386.

Views

Author

Keywords

Links

Programs

Mathematica

Formula

A026951 Self-convolution of array T given by A026386.

Views

Author

Keywords