A001793 -id:A001793 - cp's OEIS frontend

A308946 Expansion of e.g.f. 1/(1 - x(1 + x/2)exp(x)).

1, 1, 5, 30, 244, 2485, 30351, 432502, 7043660, 129050649, 2627117875, 58829021416, 1437117395946, 38032508860177, 1083932872119839, 33098858988564090, 1078083456543449416, 37309607437056658129, 1367138649165397662627, 52879280631976735387588

Offset: 0

Ilya Gutkovskiy, Jul 02 2019

nonn

Cf. A000217, A001793, A006153, A052529, A279361, A308861.

nmax = 19; CoefficientList[Series[1/(1 - x (1 + x/2) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k + 1, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

E.g.f.: 1 / (1 - Sum_{k>=1} (k*(k + 1)/2)*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000217(k) * a(n-k).
a(n) ~ n! * (2 + r) / ((2 + 4*r + r^2) * r^n), where r = 0.49122518354447387971550543450091640839121607... is the root of the equation  exp(r)*r*(2 + r) = 2. - Vaclav Kotesovec, Aug 09 2021

A209404 Negated coefficients of Chebyshev T polynomials: a(n) = -A053120(n+14, n), n >= 0.

Original entry on oeis.org

1, 15, 128, 816, 4320, 20064, 84480, 329472, 1208064, 4209920, 14057472, 45260800, 141213696, 428654592, 1270087680, 3683254272, 10478223360, 29297934336, 80648077312, 218864025600, 586290298880, 1551944908800, 4063273943040

Offset: 0

Brad Clardy, Mar 08 2012

The MAGMA program provided calculates the coefficients of order one Chebyshev polynomials, for any arbitrary level. For example, setting Rn to 0 produces A001792, 1 produces A001793, 2 produces A001794, 3 produces A006974, 4 produces A006975, and 5 produces A006976. This sequence is produced with an Rn of 6.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Cf. A001792, A001793, A001794, A006974, A006975, A006976, A053120.

List([0..30], n-> 2^(n-1)*(n+14)*Binomial(n+6,6)/7); # G. C. Greubel, Oct 18 2019

Rn:=6; [2^(n-1)/(Rn+1)*Binomial(n+Rn,Rn)*(n+(Rn+1)*2) : n in [0..22]];

R:=PowerSeriesRing(Integers(), 23); Coefficients(R!( (1-x)/(1-2*x)^8 )); // Marius A. Burtea, Oct 17 2019

seq(2^(n-1)*(n+14)*binomial(n+6,6)/7, n=0..30); # G. C. Greubel, Oct 18 2019

CoefficientList[Series[(1-x)/(1-2*x)^8, {x,0,30}], x] (* or *) Table[2^(n-1)*Binomial[n+6,6]*(n+14)/7, {n,0,30}] (* G. C. Greubel, Jan 03 2018 *)

for(n=0,30, print1(2^(n-1)*binomial(n+6,6)*(n+14)/7, ", ")) \\ G. C. Greubel, Jan 03 2018

a(n) = 2^(n-1)*binomial(n+6, 6)*(n+14)/7 = -A053120(n+14, n), n >= 0. [See a comment in A053120 on subdiagonal sequences. - Wolfdieter Lang, Jan 03 2020]
G.f.: (1-x)/(1-2*x)^8. - Colin Barker, May 31 2013
E.g.f.: (1/315)*exp(2*x)*(315 + 4095*x + 11340*x^2 + 11550*x^3 + 5250*x^4 + 1134*x^5 + 112*x^6 + 4*x^7). - Stefano Spezia, Oct 17 2019

Name made more specific by Wolfdieter Lang, Nov 25 2019

A081265 Triangle of coefficients of the polynomials a(n, x) = 2a(n-1, x)+ x^2a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 4, 0, 3, 0, 8, 0, 8, 0, 1, 16, 0, 20, 0, 5, 0, 32, 0, 48, 0, 18, 0, 1, 64, 0, 112, 0, 56, 0, 7, 0, 128, 0, 256, 0, 160, 0, 32, 0, 1, 256, 0, 576, 0, 432, 0, 120, 0, 9, 0, 512, 0, 1280, 0, 1120, 0, 400, 0, 50, 0, 1, 1024, 0, 2816, 0, 2816, 0, 1232, 0, 220

Offset: 0

Paul Barry, Mar 15 2003

Unsigned Chebyshev numbers of the first kind. Columns include A011782, A001792, A001793, A001794, A006974.

For the Riordan coefficient triangle for Chebyshev's T-polynomials (decreasing odd or even powers of x) see A039991. - Wolfdieter Lang, Aug 06 2014

			Triangle rows are {1}, {1,0}, {2,0,1}, {4,0,3,0}, {8,0,8,0,1},.... [Corrected by _Philippe Deléham_, Dec 27 2007]
See the unsigned example under A039991. - _Wolfdieter Lang_, Aug 06 2014

Cf. A008310, A039991 (signed).

T(n,k) = [x^k] a(n,x), k = 0, 1, ..., n, with polynomial a(n,x) defined by the recurrence given as name. Its Binet-de Moivre form is a(n, x) = ((1+sqrt(x^2+1))^n + (1-sqrt(x^2+1))^n)/2.

O.g.f. for row polynomials a(n,x): (1-z)/(1 - 2*z - (x*z)^2). Compare with A039991.

Edited. Name and formula clarified. G.f. of row polynomial, and crossref. A039991 added. - Wolfdieter Lang, Aug 06 2014

A124932 Triangle read by rows: T(n,k) = k(k+1)binomial(n,k)/2 (1 <= k <= n).

Original entry on oeis.org

1, 2, 3, 3, 9, 6, 4, 18, 24, 10, 5, 30, 60, 50, 15, 6, 45, 120, 150, 90, 21, 7, 63, 210, 350, 315, 147, 28, 8, 84, 336, 700, 840, 588, 224, 36, 9, 108, 504, 1260, 1890, 1764, 1008, 324, 45, 10, 135, 720, 2100, 3780, 4410, 3360, 1620, 450, 55

Offset: 1

Gary W. Adamson, Nov 12 2006

Row sums = A001793: (1, 5, 18, 56, 160, 432, ...).
Triangle is P*M, where P is the Pascal triangle as an infinite lower triangular matrix and M is an infinite bidiagonal matrix with (1,3,6,10,...) in the main diagonal and in the subdiagonal.
This number triangle can be used as a control sequence when listing combinations of subsets as in Pascals triangle by assigning a number to each element that corresponds to the n:th subset that the element belongs to. One then gets number blocks whose sums are the terms in this number triangle. - Mats Granvik, Jan 14 2009

			First few rows of the triangle:
  1;
  2,   3;
  3,   9,   6;
  4,  18,  24,  10;
  5,  30,  60,  50,  15;
  6,  45, 120, 150,  90,  21;
  7,  63, 210, 350, 315, 147,  28;
  ...
From _Mats Granvik_, Dec 18 2009: (Start)
The numbers in this triangle are sums of the following recursive number blocks:
1................................
.................................
11.....12........................
.................................
111....112....123................
.......122.......................
.................................
1111...1112...1123...1234........
.......1122...1223...............
.......1222...1233...............
.................................
11111..11112..11123..11234..12345
.......11122..11223..12234.......
.......11222..12223..12334.......
.......12222..11233..12344.......
..............12233..............
..............12333..............
.................................
(End)

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A001793.

B:=Binomial;; Flat(List([1..12], n-> List([1..n], k-> B(k+1,2)* B(n,k) ))); # G. C. Greubel, Nov 19 2019

B:=Binomial; [B(k+1,2)*B(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 19 2019

T:=(n,k)->k*(k+1)*binomial(n,k)/2: for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Table[Binomial[k + 1, 2]*Binomial[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Nov 19 2019 *)

T(n,k) = binomial(k+1,2)*binomial(n,k); \\ G. C. Greubel, Nov 19 2019

b=binomial; [[b(k+1,2)*b(n,k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 19 2019

T(n,k) = binomial(k+1,2)*binomial(n,k). - G. C. Greubel, Nov 19 2019

Edited by N. J. A. Sloane, Nov 24 2006

A125092 Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).

Original entry on oeis.org

1, 1, 4, 1, 8, 9, 1, 12, 27, 16, 1, 16, 54, 64, 25, 1, 20, 90, 160, 125, 36, 1, 24, 135, 320, 375, 216, 49, 1, 28, 189, 560, 875, 756, 343, 64, 1, 32, 252, 896, 1750, 2016, 1372, 512, 81, 1, 36, 324, 1344, 3150, 4536, 4116, 2304, 729, 100, 1, 40, 405, 1920, 5250, 9072

Offset: 0

Gary W. Adamson, Nov 19 2006

Binomial transform of the infinite diagonal matrix (1,4,9,16,...).

Sum of entries in row n = (n+1)*(n+4)*2^(n-2) = A001793(n+1).

			First few rows of the triangle:
  1;
  1,   4;
  1,   8,   9;
  1,  12,  27,  16;
  1,  16,  54,  64,  25;
  1,  20,  90, 160, 125,  36;
  ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A001793.

T:=(n,k)->(k+1)^2*binomial(n,k): for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Table[(k+1)^2 Binomial[n,k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Feb 20 2023 *)

Edited by N. J. A. Sloane, Nov 29 2006

A135294 a(n) = 3*a(n-1)+n if a(n-1) is not divisible by 2, or a(n) = a(n-1)/2 otherwise.

Original entry on oeis.org

1, 4, 2, 1, 7, 26, 13, 46, 23, 78, 39, 128, 64, 32, 16, 8, 4, 2, 1, 22, 11, 54, 27, 104, 52, 26, 13, 66, 33, 128, 64, 32, 16, 8, 4, 2, 1, 40, 20, 10, 5, 56, 28, 14, 7, 66, 33, 146, 73, 268, 134, 67, 253, 812, 406, 203, 665, 2052, 1026, 513, 1599, 4858, 2429, 7350, 3675, 11090, 5545, 16702, 8351, 25122, 12561

Offset: 1

Ctibor O. Zizka, Dec 04 2007

nonn

a(n)=a(0)*(3^(n-i))/(2^i) + c where c is in the range (0..sum(i*3^(n-i))). Sum(i*3^(n-i)) for i=1 to n equals A001793 (coefficients of Chebyshev polynomials). Max a(n) = 3^n*(a(0)/3^i*2^i + 9/4) - ((2*n+5)/4) which for large n gives max a(n) ~ 2.25*3^n - n/2. - Ctibor O. Zizka, Dec 26 2007

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A135287.

nxt[{n_,a_}]:={n+1,If[OddQ[a],3a+n,a/2]}; NestList[nxt,{1,1},70][[;;,2]] (* Harvey P. Dale, Oct 01 2024 *)

Corrected and extended by Harvey P. Dale, Oct 01 2024

A273717 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k L-shaped corners (n>=2, k>=0).

Original entry on oeis.org

1, 2, 4, 1, 8, 5, 16, 18, 1, 32, 56, 9, 64, 160, 50, 1, 128, 432, 220, 14, 256, 1120, 840, 110, 1, 512, 2816, 2912, 645, 20, 1024, 6912, 9408, 3150, 210, 1, 2048, 16640, 28800, 13552, 1575, 27, 4096, 39424, 84480, 53088, 9534, 364, 1, 8192, 92160, 239360, 193440, 49644, 3388, 35, 16384, 212992, 658944, 665280, 231000, 24822, 588

Offset: 2

Emeric Deutsch, May 29 2016

Each L-shaped corner can be viewed as a descent (as defined in Sec. 5.1 of the Blecher et al. reference). - Emeric Deutsch, Jul 02 2016

			Row 4 is 4,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] of which only [2,1] yields a |_ - shaped corner.
Triangle starts:
  1;
  2;
  4,1;
  8,5;
  16,18,1.

A. Blecher, C. Brennan, and A. Knopfmacher, Combinatorial parameters in bargraphs (preprint).

Alois P. Heinz, Rows n = 2..200, flattened
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. Appl. Math., 31, 2003, 86-112.
Emeric Deutsch and S. Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088 [math.CO], 2016.

Sum of entries in row n = A082582(n).

Cf. A001793, A273718.

eq := t*z*G^2-(1-2*z-t*z^2)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): for n from 2 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 18 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t) option remember; expand(
      `if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1))+
      `if`(t>0 or y<2, 0, b(n, y-1, -1))+
      `if`(y<1, 0, b(n-1, y, 0)*`if`(t<0, z, 1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=2..18);  # Alois P. Heinz, Jun 06 2016

b[n_, y_, t_] := b[n, y, t] = Expand[If[n == 0, 1 - t, If[t < 0, 0, b[n - 1, y + 1, 1]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1]] + If[y < 1, 0, b[n - 1, y, 0]*If[t < 0, z, 1]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0]]; Table[T[n], {n, 2, 18}] // Flatten (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

G.f.: G = G(t,z) satisfies t*z*G^2 - (1 - 2*z - t*z^2)*G + z^2 = 0.
Sum_{k>=1} k*T(n,k) = A273718(n).
T(n,0) = 2^(n-2).
T(n,1) = n*(n-3)*2^(n-6) = A001793(n-3) for n>=4.

A320905 T(n, k) = binomial(2n - 1 - k, k - 1)hypergeom([2, 2, 1-k], [1, 1 - 2k + 2n], -1), triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= n.

Original entry on oeis.org

1, 1, 5, 1, 7, 18, 1, 9, 31, 56, 1, 11, 48, 111, 160, 1, 13, 69, 198, 351, 432, 1, 15, 94, 325, 699, 1023, 1120, 1, 17, 123, 500, 1280, 2223, 2815, 2816, 1, 19, 156, 731, 2186, 4458, 6562, 7423, 6912, 1, 21, 193, 1026, 3525, 8330, 14198, 18324, 18943, 16640

Offset: 1

Peter Luschny, Oct 28 2018

			Triangle starts:
[1] 1
[2] 1,  5
[3] 1,  7,  18
[4] 1,  9,  31,  56
[5] 1, 11,  48, 111,  160
[6] 1, 13,  69, 198,  351,  432
[7] 1, 15,  94, 325,  699, 1023, 1120
[8] 1, 17, 123, 500, 1280, 2223, 2815, 2816
[9] 1, 19, 156, 731, 2186, 4458, 6562, 7423, 6912

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums with shifted indices in A318947.

T(n, n) = A001793(n).

T := (n, k) -> binomial(2*n-1-k,k-1)*hypergeom([2,2,1-k], [1,1-2*k+2*n], -1):
seq(seq(simplify(T(n, k)), k=1..n), n=1..10);

T[n_, k_] := Sum[Binomial[2*n-k, 2*n-2*k+1+j]*Binomial[j+2, 2],{j, 0, 2*n-k}]; Flatten[Table[T[n, k], {n, 1, 10}, {k, 1, n}]] (* Detlef Meya, Dec 31 2023 *)

T(n, k) = {sum(j=0, 2*n-k, binomial(2*n-k, 2*n - 2*k + 1 + j) * binomial(j+2, 2))} \\ Andrew Howroyd, Dec 31 2023

from functools import cache
@cache
def T(n, k):
    if k < 1 or n < 1: return 0
    if k == 1: return 1
    if k == n: return n * (n + 3) * 2**(n - 3)
    return T(n-1, k) + 2*T(n-1, k-1) - T(n-2, k-2)
for n in range(1, 10): print([T(n, k) for k in range(1, n+1)])
# after Detlef Meya, Peter Luschny, Jan 01 2024

T(n, k) = Sum_{j=0..2*n-k} binomial(2*n-k, 2*n - 2*k + 1 + j)*binomial(j+2, 2). - Detlef Meya, Dec 31 2023

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 2, 3, 1, 2, 3, 4, 1, 4, 2, 4, 3, 4, 1, 2, 4, 1, 3, 4, 2, 3, 4, 1, 2, 3, 4, 5, 1, 5, 2, 5, 3, 5, 4, 5, 1, 2, 5, 1, 3, 5, 1, 4, 5, 2, 3, 5, 2, 4, 5, 3, 4, 5, 1, 2, 3, 5, 1, 2, 4, 5, 1, 3, 4, 5, 2, 3, 4, 5, 1, 2, 3, 4, 5

Offset: 1

Gus Wiseman, May 11 2021

			The sets are the columns below:
  1 2 1 3 1 2 1 4 1 2 3 1 1 2 1 5 1 2 3 4 1 1 1 2 2 3 1
      2   3 3 2   4 4 4 2 3 3 2   5 5 5 5 2 3 4 3 4 4 2
              3         4 4 4 3           5 5 5 5 5 5 3
                              4                       5
As a tetrangle, the first four triangles are:
  {1}
  {2},{1,2}
  {3},{1,3},{2,3},{1,2,3}
  {4},{1,4},{2,4},{3,4},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}

Wikiversity, Lexicographic and colexicographic order

Triangle lengths are A000079.
Triangle sums are A001793.
Positions of first appearances are A005183.
Set maxima are A070939.
Set lengths are A124736.
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A296774, A299755, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A036043, A049085, A115623, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Mathematica
```
SortBy[Rest[Subsets[Range[5]]],Last]
```

A069515 Number of transpositions (interchanges of adjacent digits, sometimes called inversions) needed to change all n-digit base 3 numbers into nondecreasing order.

Original entry on oeis.org

0, 3, 24, 135, 648, 2835, 11664, 45927, 174960, 649539, 2361960, 8444007, 29760696, 103630995, 357128352, 1219657095, 4132485216, 13904090883, 46490458680, 154580775111, 511395045480, 1684116865683, 5523066491184

Offset: 1

John W. Layman, Apr 16 2002

The corresponding problem for base 2 numbers gives a(n)=A001793(n-1) for n=2,3,4,....

			The base 3 number 1210 requires 4 transpositions: 1210->1201->1021->0121->0112.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A064017.

LinearRecurrence[{9, -27, 27}, {0, 3, 24}, 50] (* Paolo Xausa, Mar 18 2024 *)
nxt[{n_,a_}]:={n+1,3a+(2n+1)3^(n-1)}; NestList[nxt,{1,0},30][[;;,2]] (* Harvey P. Dale, Aug 23 2024 *)

a(n) = 3a(n-1)+(2n-1)3^(n-2).
a(n) = (n-1)(n+1)3^(n-2). - Ralf Stephan, Sep 02 2003
G.f.: 3*x^2*(1-x)/(1-3*x)^3. - Colin Barker, May 01 2012

Corrected by T. D. Noe, Nov 01 2006

cp's OEIS Frontend

A308946 Expansion of e.g.f. 1/(1 - x*(1 + x/2)*exp(x)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A209404 Negated coefficients of Chebyshev T polynomials: a(n) = -A053120(n+14, n), n >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

Formula

Extensions

A081265 Triangle of coefficients of the polynomials a(n, x) = 2*a(n-1, x)+ x^2*a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A124932 Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A125092 Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A135294 a(n) = 3*a(n-1)+n if a(n-1) is not divisible by 2, or a(n) = a(n-1)/2 otherwise.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A273717 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k L-shaped corners (n>=2, k>=0).

Views

Author

Keywords

Comments

Examples

References

A308946 Expansion of e.g.f. 1/(1 - x(1 + x/2)exp(x)).

A081265 Triangle of coefficients of the polynomials a(n, x) = 2a(n-1, x)+ x^2a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.

A124932 Triangle read by rows: T(n,k) = k(k+1)binomial(n,k)/2 (1 <= k <= n).

A320905 T(n, k) = binomial(2n - 1 - k, k - 1)hypergeom([2, 2, 1-k], [1, 1 - 2k + 2n], -1), triangle read by rows, T(n, k) for n >= 1 and 1 <= k <= n.