A060099 -id:A060099 - cp's OEIS frontend

A096338 a(n) = (2/(n-1))a(n-1) + ((n+5)/(n-1))a(n-2) with a(0)=0 and a(1)=1.

0, 1, 2, 6, 10, 20, 30, 50, 70, 105, 140, 196, 252, 336, 420, 540, 660, 825, 990, 1210, 1430, 1716, 2002, 2366, 2730, 3185, 3640, 4200, 4760, 5440, 6120, 6936, 7752, 8721, 9690, 10830, 11970, 13300, 14630, 16170, 17710, 19481, 21252, 23276, 25300, 27600

Offset: 0

Benoit Cloitre, Jun 28 2004

Without the leading zero, Poincaré series [or Poincare series] P(C_{2,2}; t).
Starting (1, 2, 6, ...) = partial sums of the tetrahedral numbers, A000292 with repeats: (1, 1, 4, 4, 10, 10, 20, 20, 35, 35, ...). - Gary W. Adamson, Mar 30 2009
Starting with 1 = [1, 2, 3, ...] convolved with the aerated triangular series, [1, 0, 3, 0, 6, ...]. - Gary W. Adamson, Jun 11 2009
From Alford Arnold, Oct 14 2009: (Start)
a(n) is also related to Dyck Paths. Note that
0 1 2 6 10 20 30 50 70 105 ...
minus
0 0 0 0 1 2 6 10 20 30 ...
equals
0 1 2 6 9 18 24 40 50 75 ... A028724
(End)
The Kn11, Kn12, Kn13, Fi1 and Ze1 triangle sums of A139600 are related to the sequence given above; e.g., Ze1(n) = 3*A096338(n-1) - 3*A096338(n-3) + 9*A096338(n-4), with A096338(n) = 0 for n <= -1. For the definition of these triangle sums, see A180662. - Johannes W. Meijer, Apr 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Dragomir Z. Djokovic, Poincaré series [or Poincare series] of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. See Table 3.
Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 3.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4, 19.
M. Navascues and T. Vertesi, The Structure of Matrix Product States, arXiv preprint arXiv:1509.04507 [quant-ph], 2015-2018.
Miguel Navascués and Tamás Vértesi, Bond dimension witnesses and the structure of homogeneous matrix product states, Quantum 2 (2018): 50.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A006918, A038163, A060099, A058187, A000292, A028724.

A096338:=n->-(floor(n/2)+1)*(floor(n/2)+2)*(floor(n/2)+3)*(3*floor(n/2)-2*n)/12; seq(A096338(k),k=0..100); # Wesley Ivan Hurt, Oct 04 2013

t = {0, 1}; Do[AppendTo[t, (2/(n - 1))*t[[-1]] + ((n + 5)/(n - 1))*t[[-2]]], {n, 2, 50}]; t (* T. D. Noe, Oct 08 2013 *)
CoefficientList[Series[x/((1 - x)^2*(1 - x^2)^3), {x, 0, 45}], x] (* or *)
Nest[Append[#1, (2/(#2 - 1))*#1[[-1]] + ((#2 + 5)/(#2 - 1))*#1[[-2]] ] & @@ {#, Length@ #} &, {0, 1}, 44] (* Michael De Vlieger, May 30 2018 *)

G.f.: x/((1-x)^2*(1-x^2)^3). - Ralf Stephan, Jun 29 2004
a(n) = Sum_{k=1..floor(n/2)+1} ( Sum_{i=1..k} i*(n-2*k+2) ) = -(floor(n/2)+1) * (floor(n/2)+2) * (floor(n/2)+3) * (3*floor(n/2) - 2*n)/12. - Wesley Ivan Hurt, Sep 26 2013
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8). - Wesley Ivan Hurt, Nov 26 2020
128*a(n) = 8*n^3 +94/3*n^2 +44*n +15 +2/3*n^4 -2*(-1)^n*n^2 -12*(-1)^n*n -15*(-1)^n. - R. J. Mathar, Mar 23 2021

A060098 Triangle of partial sums of column sequences of triangle A060086, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 9, 16, 13, 5, 1, 1, 12, 30, 32, 19, 6, 1, 1, 16, 50, 71, 55, 26, 7, 1, 1, 20, 80, 140, 140, 86, 34, 8, 1, 1, 25, 120, 259, 316, 246, 126, 43, 9, 1, 1, 30, 175, 448, 660, 622, 399, 176, 53, 10, 1

Offset: 0

Wolfdieter Lang, Apr 06 2001

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The g.f. for the row polynomials p(n,x) = Sum_{m=0..n} a(n,m)*x^m is (1/(1-x*z/((1-z^2)*(1-z))))/(1-z).
Row sums give A052534. Column sequences (without leading zeros) give A000012 (powers of 1), A002620(n+1), A002624, A060099-A060101 for m=0..5.
The bisections of the column sequences give triangles A060102 and A060556.
Riordan array (1/(1-x), x/((1-x)*(1-x^2))). - Paul Barry, Mar 28 2011

			p(3,x) = 1 + 4*x + 3*x^2 + x^3.
Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  3,  1;
  1,  6,  8,  4,  1;
  1,  9, 16, 13,  5,  1;
  1, 12, 30, 32, 19,  6,  1;
  1, 16, 50, 71, 55, 26,  7,  1;
  ...

Vincenzo Librandi, Rows n = 0..100, flattened
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.

Cf. A052534, A000012, A002620(n+1), A002624, A060099, A060100, A060101.

Cf. A060102, A060556, A188316.

A060098 := proc(n,k) add( binomial(n-2*i,n-2*i-k)*binomial(k+i-1,i), i=0..floor(n/2)) ; end proc:
seq(seq(A060098(n,k), k=0..n), n=0..12); # R. J. Mathar, Mar 29 2011
# Recurrence after Philippe Deléham:
T := proc(n, k) option remember;
if k < 0 or k > n then 0 elif k = 0 or n = k then 1 else
T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, May 07 2023

t[n_, k_] := Sum[ Binomial[n-2*j, n-2*j-k]*Binomial[k+j-1, j], {j, 0, n/2}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

G.f. for column m >= 0: ((x/((1-x^2)*(1-x)))^m)/(1-x) = x^m/((1+x)^m*(1-x)^(2*m+1)).
Number triangle T(n,k) = Sum_{i=0..floor(n/2)} C(n-2*i,n-2*i-k)*C(k+i-1,i). - Paul Barry, Mar 28 2011
From Philippe Deléham, Apr 20 2023: (Start)
T(n, k) = 0 if k < 0 or if k > n; T(n, k) = 1 if k = 0 or k = n; otherwise:
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-3, k).
T(n, k) = A188316(n, k) + A188316(n-1, k). (End)

A018210 Alkane (or paraffin) numbers l(9,n).

Original entry on oeis.org

1, 4, 16, 44, 110, 236, 472, 868, 1519, 2520, 4032, 6216, 9324, 13608, 19440, 27192, 37389, 50556, 67408, 88660, 115258, 148148, 188552, 237692, 297115, 368368, 453376, 554064, 672792, 811920, 974304, 1162800, 1380825, 1631796

Offset: 0

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

From M. F. Hasler, May 02 2009: (Start)
Also, 6th column of A159916, i.e., number of 6-element subsets of {1,...,n+6} whose elements add up to an odd integer.
Third differences are A002412([n/2]). (End)
F(1,6,n) is the number of bracelets with 1 blue, 6 identical red and n identical black beads. If F(1,6,1) = 4 and F(1,6,2) = 16 taken as a base, F(1,6,n) = n(n+1)(n+2)(n+3)(n+4)/120 + F(1,4,n) + F(1,6,n-2). F(1,4,n) is the number of bracelets with 1 blue, 4 identical red and n identical black beads. If F(1,4,1) = 3 and F(1,4,2) = 9 taken as a base; F(1,4,n) = n(n+1)(n+2)/6 + F(1,2,n) + F(1,4,n-2). F(1,2,n) is the number of bracelets with 1 blue, 2 identical red and n identical black beads. If F(1,2,1) = 2 and F(1,2,2) = 4 taken as a base F(1,2,n) = n + 1 + F(1,2,n-2). - Ata Aydin Uslu and Hamdi G. Ozmenekse, Mar 16 2012

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
Winston C. Yang (paper in preparation).

N. J. A. Sloane, Classic Sequences
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Ata A. Uslu and Hamdi G. Ozmenekse, F(1,6,n)
Ata A. Uslu and Hamdi G. Ozmenekse, F(1,4,n)
Ata A. Uslu and Hamdi G. Ozmenekse, F(1,2,n)
Index entries for linear recurrences with constant coefficients, signature (4, -3, -8, 14, 0, -14, 8, 3, -4, 1).

Cf. A005995 (first differences).

a:=n-> (Matrix([[1,0$7,3,12]]). Matrix(10, (i,j)-> if (i=j-1) then 1 elif j=1 then [4, -3, -8, 14, 0, -14, 8, 3, -4, 1][i] else 0 fi)^n)[1,1]: seq (a(n), n=0..33); # Alois P. Heinz, Jul 31 2008

CoefficientList[(1+3*x^2)/((1-x)^7*(1+x)^3) + O[x]^34, x] (* Jean-François Alcover, Jun 08 2015 *)
LinearRecurrence[{4, -3, -8, 14, 0, -14, 8, 3, -4, 1},{1, 4, 16, 44, 110, 236, 472, 868, 1519, 2520},34] (* Ray Chandler, Sep 23 2015 *)

A018210(n)=(n+2)*(n+4)*(n+6)^2*(n^2+3*n+5)/1440-if(n%2,(n^2+7*n+11)/32) \\ M. F. Hasler, May 02 2009

G.f.: (1+3*x^2)/(1-x)^4/(1-x^2)^3. - N. J. A. Sloane
l(c, r) = 1/2 C(c+r-3, r) + 1/2 d(c, r), where d(c, r) is C((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, C((c + r - 4)/2, r/2) if c is even and r is even, C((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.
a(2n) = (n+1)(n+2)(n+3)^2(4n^2+6n+5)/90, a(2n-1) = n(n+1)(n+2)(n+3)(4n^2+6n+5)/90. - M. F. Hasler, May 02 2009
a(n) = (1/(2*6!))*(n+2)*(n+4)*(n+6)*((n+1)*(n+3)*(n+5) + 1*3*5) - (1/2)*(1/2^4)*(n^2+7*n+11)*(1/2)*(1-(-1)^n). - Yosu Yurramendi, Jun 23 2013
a(n) = A060099(n)+3*A060099(n-2). - R. J. Mathar, May 08 2020

A299337 Expansion of 1 / ((1 - x)^7*(1 + x)^5).

Original entry on oeis.org

1, 2, 8, 14, 35, 56, 112, 168, 294, 420, 672, 924, 1386, 1848, 2640, 3432, 4719, 6006, 8008, 10010, 13013, 16016, 20384, 24752, 30940, 37128, 45696, 54264, 65892, 77520, 93024, 108528, 128877, 149226, 175560, 201894, 235543, 269192, 311696, 354200, 407330

Offset: 0

Colin Barker, Feb 07 2018

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).

Cf. A001769, A060099, A299335, A299336, A299338.

LinearRecurrence[{2, 4, -10, -5, 20, 0, -20, 5, 10, -4, -2, 1}, {1, 2, 8, 14, 35, 56, 112, 168, 294, 420, 672, 924}, 41] (* Michael De Vlieger, Dec 19 2024 *)

Vec(1 / ((1 - x)^7*(1 + x)^5) + O(x^40))

a(n) = (2*n^6 + 72*n^5 + 1040*n^4 + 7680*n^3 + 30368*n^2 + 60288*n + 46080) / 46080 for n even.
a(n) = (2*n^6 + 72*n^5 + 1010*n^4 + 6960*n^3 + 24278*n^2 + 39048*n + 20790) / 46080 for n odd.
a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n>11.

A299335 Expansion of 1 / ((1 - x)^7*(1 + x)^2).

Original entry on oeis.org

1, 5, 17, 45, 103, 211, 399, 707, 1190, 1918, 2982, 4494, 6594, 9450, 13266, 18282, 24783, 33099, 43615, 56771, 73073, 93093, 117481, 146965, 182364, 224588, 274652, 333676, 402900, 483684, 577524, 686052, 811053, 954465, 1118397, 1305129, 1517131, 1757063

Offset: 0

Colin Barker, Feb 07 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,0,14,-14,0,8,-5,1).

Cf. A001769, A060099, A299336, A299337, A299338.

CoefficientList[Series[1/((1 - x)^7 (1 + x)^2), {x, 0, 40}], x] (* or *) LinearRecurrence[{5, -8, 0, 14, -14, 0, 8, -5, 1}, {1, 5, 17, 45, 103, 211, 399, 707, 1190}, 41] (* Robert G. Wilson v, Feb 07 2018 *)

Vec(1 / ((1 - x)^7*(1 + x)^2) + O(x^40))

a(n) = (2*n^6 + 54*n^5 + 575*n^4 + 3060*n^3 + 8468*n^2 + 11376*n + 5760) / 5760 for n even.
a(n) = (2*n^6 + 54*n^5 + 575*n^4 + 3060*n^3 + 8468*n^2 + 11286*n + 5355) / 5760 for n odd.
a(n) = 5*a(n-1) - 8*a(n-2) + 14*a(n-4) - 14*a(n-5) + 8*a(n-7) - 5*a(n-8) + a(n-9) for n>8.

A299336 Expansion of 1 / ((1 - x)^7*(1 + x)^4).

Original entry on oeis.org

1, 3, 10, 22, 49, 91, 168, 280, 462, 714, 1092, 1596, 2310, 3234, 4488, 6072, 8151, 10725, 14014, 18018, 23023, 29029, 36400, 45136, 55692, 68068, 82824, 99960, 120156, 143412, 170544, 201552, 237405, 278103, 324786, 377454, 437437, 504735, 580888, 665896

Offset: 0

Colin Barker, Feb 07 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).

Cf. A001769, A060099, A299335, A299337, A299338.

Vec(1 / ((1 - x)^7*(1 + x)^4) + O(x^40))

a(n) = (2*n^6 + 66*n^5 + 860*n^4 + 5640*n^3 + 19568*n^2 + 33984*n + 23040) / 23040 for n even.
a(n) = (2*n^6 + 66*n^5 + 860*n^4 + 5580*n^3 + 18578*n^2 + 28914*n + 15120) / 23040 for n odd.
a(n) = 3*a(n-1) + a(n-2) - 11*a(n-3) + 6*a(n-4) + 14*a(n-5) - 14*a(n-6) - 6*a(n-7) + 11*a(n-8) - a(n-9) - 3*a(n-10) + a(n-11) for n>10.

A299338 Expansion of 1 / ((1 - x)^7*(1 + x)^6).

Original entry on oeis.org

1, 1, 7, 7, 28, 28, 84, 84, 210, 210, 462, 462, 924, 924, 1716, 1716, 3003, 3003, 5005, 5005, 8008, 8008, 12376, 12376, 18564, 18564, 27132, 27132, 38760, 38760, 54264, 54264, 74613, 74613, 100947, 100947, 134596, 134596, 177100, 177100, 230230, 230230

Offset: 0

Colin Barker, Feb 07 2018

Same as A000579 but with repeated terms.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).

Cf. A000579, A001769, A060099, A299335, A299336, A299337.

CoefficientList[Series[1/((1-x)^7(1+x)^6),{x,0,50}],x] (* or *) LinearRecurrence[ {1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{1,1,7,7,28,28,84,84,210,210,462,462,924},50] (* Harvey P. Dale, Oct 09 2018 *)

Vec(1 / ((1 - x)^7*(1 + x)^6) + O(x^40))

a(n) = (2*n^6 + 84*n^5 + 1400*n^4 + 11760*n^3 + 51968*n^2 + 112896*n + 92160) / 92160 for n even.
a(n) = (2*n^6 + 72*n^5 + 1010*n^4 + 6960*n^3 + 24278*n^2 + 39048*n + 20790) / 92160 for n odd.
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13) for n>12.

cp's OEIS Frontend

A096338 a(n) = (2/(n-1))*a(n-1) + ((n+5)/(n-1))*a(n-2) with a(0)=0 and a(1)=1.

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A060098 Triangle of partial sums of column sequences of triangle A060086, read by rows.

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A018210 Alkane (or paraffin) numbers l(9,n).

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A299337 Expansion of 1 / ((1 - x)^7*(1 + x)^5).

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A299335 Expansion of 1 / ((1 - x)^7*(1 + x)^2).

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A299336 Expansion of 1 / ((1 - x)^7*(1 + x)^4).

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A299338 Expansion of 1 / ((1 - x)^7*(1 + x)^6).

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A096338 a(n) = (2/(n-1))a(n-1) + ((n+5)/(n-1))a(n-2) with a(0)=0 and a(1)=1.