A008620 -id:A008620 - cp's OEIS frontend

A118923 Triangle T(n,k) built by placing T(n,0)=A000012(n) in the left edge, T(n,n)=A079978(n) on the right edge and filling the body with the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 3, 3, 2, 0, 1, 4, 6, 5, 2, 0, 1, 5, 10, 11, 7, 2, 1, 1, 6, 15, 21, 18, 9, 3, 0, 1, 7, 21, 36, 39, 27, 12, 3, 0, 1, 8, 28, 57, 75, 66, 39, 15, 3, 1, 1, 9, 36, 85, 132, 141, 105, 54, 18, 4, 0, 1, 10, 45, 121, 217, 273, 246, 159, 72, 22, 4, 0, 1, 11, 55, 166

Offset: 0

Alford Arnold, May 05 2006

The fourth diagonal is 1, 2, 5, 11, 21, ..., which is 1 + A000292. The fifth diagonal is 0, 2, 7, 18, 39, 75, 132, 217, 338, 504, 725, 1012, ..., which is A051743.
The array A007318 is generated by placing A000012 on both edges with the same Pascal-like recurrence, and the array A059259 uses edges defined by A000012 and A059841. - R. J. Mathar, Jan 21 2008
From Michael A. Allen, Nov 30 2021: (Start)
T(n,n-k) is the (n,k)-th entry of the (1/(1-x^3), x/(1-x)) Riordan array.
Sums of rows give A077947.
Sums of antidiagonals give A079962. (End)

			The table begins
  1
  1  0
  1  1  0
  1  2  1  1
  1  3  3  2  0
  1  4  6  5  2  0
  1  5 10 11  7  2  1
  1  6 15 21 18  9  3  0

Cf. A000292, A079978, A008620, A079998, A051743, A077947.

Generalization of A007318, A059259.

A000012 := proc(n) 1 ; end: A079978 := proc(n) if n mod 3 = 0 then 1; else 0 ; fi ; end: A118923 := proc(n,k) if k = 0 then A000012(n); elif k = n then A079978(n) ; else A118923(n-1,k)+A118923(n-1,k-1) ; fi ; end: for n from 0 to 15 do for k from 0 to n do printf("%d, ",A118923(n,k)) ; od: od: # R. J. Mathar, Jan 21 2008

Flatten@Table[CoefficientList[Series[1/((1 + x*y + x^2*y^2)(1 - x - x*y)), {x, 0, 23}, {y, 0, 11}], {x, y}][[n + 1, k + 1]], {n, 0, 11}, {k, 0, n}] (* Michael A. Allen, Nov 30 2021 *)

From Michael A. Allen, Nov 30 2021: (Start)
For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/3)} binomial(n-3*j,n-k)/(n-3*j).
G.f.: 1/((1+x*y+(x*y)^2)*(1-x-x*y)). (End)

Edited and extended by R. J. Mathar, Jan 21 2008

Offset changed by Michael A. Allen, Nov 30 2021

A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 3, 0, 1, 0, 1, 1, 0, 2, 0, 3, 1, 1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 4, 1, 1, 0, 1, 1, 0, 1, 2, 0, 3, 0, 5, 0, 1, 0, 1, 1, 2, 0, 1, 2, 0, 3, 0, 5, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 02 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1, 1, 1,  1, ...
  0, 1, 0, 1, 0, 1, 0, 1, 0, 0,  1, 1, 0, 0,  1, ...
  0, 1, 1, 2, 0, 1, 0, 1, 1, 0,  2, 1, 1, 0,  2, ...
  0, 1, 0, 2, 1, 2, 0, 1, 1, 0,  3, 1, 0, 0,  2, ...
  0, 1, 1, 3, 0, 2, 1, 2, 1, 0,  4, 1, 2, 0,  4, ...
  0, 1, 0, 3, 0, 2, 0, 2, 1, 1,  5, 2, 0, 0,  4, ...
  0, 1, 1, 4, 1, 3, 0, 2, 2, 0,  7, 2, 2, 1,  6, ...
  0, 1, 0, 4, 0, 3, 0, 2, 1, 0,  8, 2, 0, 0,  6, ...
  0, 1, 1, 5, 0, 3, 1, 3, 2, 0, 10, 2, 3, 0,  9, ...
  0, 1, 0, 5, 1, 4, 0, 3, 2, 0, 12, 2, 0, 0,  9, ...
  0, 1, 1, 6, 0, 4, 0, 3, 2, 1, 14, 3, 3, 0, 12, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-11, 13-20, 23 give: A000007, A000012, A059841, A008619, A079978, A008620, A121262, A008621, A103221, A079998, A001399, A002266(n+5), A079979, A008642, A097992, A008616, A008679, A082784, A000115, A025767, A008676.
Main diagonal gives A246721.
Cf. A246688, A246690 (the same for compositions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1, `if`(l=[], 0,
      add(g(n-l[-1]*j, subsop(-1=NULL, l)), j=0..n/l[-1])))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..16);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i > n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[ n >= Length[l], l = Join[l, b[i, 1]]; i++ ]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n == 0, 1, If[l == {}, 0, Sum[g[n - l[[-1]] j, ReplacePart[l, -1 -> Nothing]], {j, 0, n/l[[-1]]}]]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 16}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A029047 Expansion of 1/((1-x)(1-x^3)(1-x^6)*(1-x^10)).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 4, 4, 6, 7, 7, 10, 11, 11, 14, 16, 16, 20, 22, 23, 27, 30, 31, 36, 39, 41, 46, 50, 52, 59, 63, 66, 73, 78, 81, 90, 95, 99, 108, 115, 119, 130, 137, 142, 153, 162, 167, 180, 189, 196, 209, 220, 227, 242, 253, 262, 277, 290, 299, 317, 330

Offset: 0

N. J. A. Sloane

Number of partitions of n into the first four triangular numbers, 1, 3, 6 and 10.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,0,1,-1,0,-1,2,-1,0,-1,1,0,-1,1,0,1,-1).

Cf. A008620.

M:= Matrix(20, (i,j)-> if (i=j-1) or (j=1 and member(i, [1, 3, 6, 14, 17, 19])) then 1 elif j=1 and member(i, [4, 7, 9, 11, 13, 16, 20]) then -1 elif j=1 and i=10 then 2 else 0 fi): a:= n-> (M^(n))[1,1]: seq(a(n), n=0..80); # Alois P. Heinz, Jul 25 2008

CoefficientList[Series[1/((1-x)(1-x^3)(1-x^6)(1-x^10)),{x,0,70}],x] (* Harvey P. Dale, Feb 06 2020 *)

Vec(1/((1-x)*(1-x^3)*(1-x^6)*(1-x^10))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

a(n)=floor((2*n^3+60*n^2+527*n+1243+9*(n+1)*(-1)^n+(120*(n\3+1)*[1,1,-2]+20*[61,41,0])[n%3+1])/2160) \\ Tani Akinari, May 07 2014

a(n) = floor((2*n^3 + 60*n^2 + 567*n + 9*n*(-1)^n + 2160)/2160 - (n/18)*[(n mod 3)=2] + (1/5)*([(n mod 6)=0] - [(n mod 6)=5])). - Hoang Xuan Thanh, Jul 06 2025

A129920 Expansion of -1/(1 - x + 3x^2 - 2x^3 + x^4 - 2*x^5 + x^6).

Original entry on oeis.org

-1, -1, 2, 3, -4, -10, 5, 29, 2, -76, -45, 178, 212, -361, -750, 565, 2282, -306, -6206, -2428, 15176, 14353, -32719, -55104, 57933, 176234, -61524, -499047, -97429, 1271400, 921652, -2887641, -3948938, 5590078, 13380187, -7828378, -39536779, 108416, 104810904

Offset: 0

Roger L. Bagula, Jun 05 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-3,2,-1,2,-1).

Cf. A008620, A010892, A014019, A099443, A099479, A099480, A112712.

Cf. A125629, A129704, A129903.

R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( -1/(1-x+3*x^2-2*x^3+x^4-2*x^5+x^6) )); // G. C. Greubel, Sep 28 2024

CoefficientList[Series[-1/(1-x +3*x^2 -2*x^3 +x^4 -2*x^5 +x^6), {x,0,50}], x]

def A129920_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( -1/(1-x+3*x^2-2*x^3+x^4-2*x^5+x^6) ).list()
A129920_list(50) # G. C. Greubel, Sep 28 2024

a(n) = a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6), n >= 6. - Franck Maminirina Ramaharo, Jan 08 2019

Edited by Franck Maminirina Ramaharo, Jan 08 2019

A141828 a(n) = (n^4*a(n-1)-1)/(n-1) for n >= 2, with a(0) = 1, a(1) = 5.

Original entry on oeis.org

1, 5, 79, 3199, 272981, 42653281, 11055730435, 4424134795739, 2588750874763849, 2123099311165701661, 2358999234628557401111, 3453810779419670890966615, 6510747302004208690462157149, 15496121141045183700690805861049

Offset: 0

Peter Bala, Jul 09 2008

For related recurrences of the form a(n) = (n^k*a(n-1)-1)/(n-1) see A001339, A007808 (both k = 2) and A141827 (k = 3). a(n) is a difference divisibility sequence, that is, the difference a(n) - a(m) is divisible by n - m for all n and m (provided n is not equal to m). See A000522 for further properties of difference divisibility sequences.

Harvey P. Dale, Table of n, a(n) for n = 0..181

Cf. A001339, A007808, A141827.

a := n -> n!^3*add((n-k+1)*(k^2+k+1)/k!^3, k = 0..n): seq(a(n), n = 0..16);

nxt[{n_,a_}]:={n+1,((n+1)^4*a-1)/n}; Join[{1},NestList[nxt,{1,5},15][[All,2]]] (* Harvey P. Dale, Mar 12 2017 *)

Sum_{n>=0} a(n)*x^n/n!^3 = (1/(1-x)^2)*Sum_{n>=0} (n^2+n+1)*x^n/n!^3.
a(n) = n!^3*Sum_{k=0..n} (n-k+1)*(k^2+k+1)/k!^3.
a(n) = n*n!^3*(5 - Sum_{k=2..n} 1/(k!^3*k*(k-1))) for n > 0. [corrected by Jason Yuen, Jan 31 2025]
Congruence property: a(n) == (1+n+n^2+n^3) (mod n^4).
The recurrence a(n) = (n^3+n^2+n+2)*a(n-1) - (n-1)^3*a(n-2), n >= 2, shows that a(n) is always a positive integer. The sequence b(n) := n*n!^3 also satisfies the same recurrence with b(0) = 0, b(1) = 1. Hence we obtain the finite continued fraction expansion a(n)/(n*n!^3) = 5 - 1^3/(16 - 2^3/(41 - 3^3/(86 -...-(n-1)^3/(n^3+n^2+n+2)))), for n >= 1. a(n)*b(n+1) - b(n)*a(n+1) = n!^3.
Limit_{n->oo} a(n)/(n*n!^3) = Sum_{n>=0} (n^2+n+1)/n!^3 = 4.9367223378... .
Limit_{n->oo} a(n)/(n*n!^3) = 1 + Sum_{n>=0} 1/(Product_{k=0..n} A008620(k)).

A331943 a(n) = n^2 + 1 - ceiling((n + 2)/3).

Original entry on oeis.org

1, 3, 8, 15, 23, 34, 47, 61, 78, 97, 117, 140, 165, 191, 220, 251, 283, 318, 355, 393, 434, 477, 521, 568, 617, 667, 720, 775, 831, 890, 951, 1013, 1078, 1145, 1213, 1284, 1357, 1431, 1508, 1587, 1667, 1750, 1835, 1921, 2010, 2101, 2193, 2288, 2385, 2483, 2584

Offset: 1

Hugo Pfoertner, Feb 10 2020

Related to expansion of exp(2*(H_k-gamma))/k^2 in powers of 1/k as given by A331777/A331778.

The agreement with the results of the PARI code needs an explanation. All numerators corresponding to the computed denominators are 1.

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

Cf. A001008, A001620, A002805, A008620, A331777, A331778.

Table[n^2+1-Ceiling[(n+2)/3],{n,60}] (* or *) LinearRecurrence[{2,-1,1,-2,1},{1,3,8,15,23},60] (* Harvey P. Dale, Aug 30 2021 *)

H(n)=sum(j=1,n,1/j);
A(k)=exp(2*(H(k)-Euler))/k^2;
for(k=1,51,r=(1/k)*(A(k)-1);print1(denominator(bestappr(r,k*k)),", "))

From Colin Barker, Feb 10 2020: (Start)
G.f.: x*(1 + x + 3*x^2 + x^3) / ((1 - x)^3*(1 + x + x^2)).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>5.
(End)
E.g.f.: (1/9)*(3*exp(x)*x*(2 + 3*x) + 2*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2)). - Stefano Spezia, Feb 14 2020

A348388 Irregular triangle read by rows: T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 5, 2, 1, 6, 2, 1, 7, 3, 1, 1, 8, 3, 2, 1, 9, 4, 2, 1, 1, 10, 4, 2, 1, 1, 11, 5, 3, 2, 1, 1, 12, 5, 3, 2, 1, 1, 13, 6, 3, 2, 1, 1, 1, 14, 6, 4, 2, 2, 1, 1, 15, 7, 4, 3, 2, 1, 1, 1, 16, 7, 4, 3, 2, 1, 1, 1, 17, 8, 5, 3, 2, 2, 1, 1, 1, 18, 8, 5, 3, 2, 2, 1, 1, 1, 19, 9, 5, 4, 3, 2, 1, 1, 1, 1

Offset: 2

Wolfdieter Lang, Oct 31 2021

This irregular triangle T(n, k) gives the number of multiples of number k, larger than k and not exceeding n, for k = 1, 2, ..., floor(n/2), for n >= 2. See A348389 for the array of these multiples.
The length of row n is floor(n/2) = A004526(n), for n >= 2.
The row sums give A002541(n). See the formula given there by Wesley Ivan Hurt, May 08 2016.
The columns give the k-fold repeated positive integers k, for k >= 1.

			The irregular triangle T(n, k) begins:
n\k   1 2 3 4 5 6 7 8 9 10 ...
------------------------------
2:    1
3:    2
4:    3 1
5:    4 1
6:    5 2 1
7:    6 2 1
8:    7 3 1 1
9:    8 3 2 1
10:   9 4 2 1 1
11:  10 4 2 1 1
12:  11 5 3 2 1 1
13:  12 5 3 2 1 1
14:  13 6 3 2 1 1 1
15:  14 6 4 2 2 1 1
16:  15 7 4 3 2 1 1 1
17:  16 7 4 3 2 1 1 1
18:  17 8 5 3 2 2 1 1 1
19:  18 8 5 3 2 2 1 1 1
20:  19 9 5 4 3 2 1 1 1  1
...

Karl-Heinz Hofmann, Table of n, a(n) for n = 2..10101

Columns k (with varying offsets): A000027, A004526, A008620, A008621, A002266, A097992, ...

Cf. A002541, A004526, A010766, A348389.

T[n_, k_] := Floor[(n - k)/k]; Table[T[n, k], {n, 2, 20}, {k, 1, Floor[n/2]}] // Flatten (* Amiram Eldar, Nov 02 2021 *)

def A348388row(n): return [(n - k) // k for k in range(1, 1 + n // 2)]
for n in range(2, 21): print(A348388row(n))  # Peter Luschny, Nov 05 2021

T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

G.f. of column k: G(k, x) = x^(2*k)/((1 - x)*(1 - x^k)).

A353205 Expansion of e.g.f. (1 - x^3)^(1 + 1/x + 1/x^2).

Original entry on oeis.org

1, -1, -1, -1, 13, 19, -29, 251, 281, -13033, 56071, -28601, -10136411, 57321419, -39757717, -17223709021, 139901102641, -12418205969, -56710054724849, 628073178260687, 380303328920381, -324513582131326141, 4616335903275095539, 5642278545451902859

Offset: 0

Seiichi Manyama, Apr 30 2022

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..449

Cf. A008620, A246689, A353204.

nmax = 25; CoefficientList[Series[(1 - x^3)^(1 + 1/x + 1/x^2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 09 2022 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3)^(1+1/x+1/x^2)))

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, x^k/((k+2)\3)))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!*sum(j=1, i, j/((j+2)\3)*v[i-j+1]/(i-j)!)); v;

E.g.f.: exp( -Sum{k >= 1} x^k/A008620(k-1) ).

a(0) = 1; a(n) = -(n-1)! * Sum_{k=1..n} k/A008620(k-1) * a(n-k)/(n-k)!.

A052569 E.g.f. 1/((1-x)(1-x^3)).

Original entry on oeis.org

1, 1, 2, 12, 48, 240, 2160, 15120, 120960, 1451520, 14515200, 159667200, 2395008000, 31135104000, 435891456000, 7846046208000, 125536739328000, 2134124568576000, 44816615940096000, 851515702861824000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Robert Israel, Table of n, a(n) for n = 0..448
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 512

spec := [S,{S=Prod(Sequence(Prod(Z,Z,Z)),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# Alternative:
f:= gfun:-rectoproc({ a(1)=1, a(0)=1, a(2)=2, (-14*n-n^3-7*n^2-8)*a(n)+(-2-n)*a(n+1)+a(n+3)-a(n+2)=0},a(n),remember):
map(f, [$0..30]); # Robert Israel, Sep 25 2019

With[{nn=20},CoefficientList[Series[1/((1-x)(1-x^3)),{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Aug 25 2012 *)

E.g.f.: 1/(-1+x)/(-1+x^3)
Recurrence: {a(1)=1, a(0)=1, a(2)=2, (-14*n-n^3-7*n^2-8)*a(n)+(-2-n)*a(n+1)+a(n+3)-a(n+2)=0}
(1/3*n+2/3+Sum(1/9*(-1+_alpha)*_alpha^(-1-n), _alpha=RootOf(_Z^2+_Z+1)))*n!
a(n) = n!*A008620(n). - R. J. Mathar, Nov 27 2011

A058936 Decomposition of Stirling's S(n,2) based on associated numeric partitions.

Original entry on oeis.org

0, 1, 3, 8, 3, 30, 20, 144, 90, 40, 840, 504, 420, 5760, 3360, 2688, 1260, 45360, 25920, 20160, 18144, 403200, 226800, 172800, 151200, 72576, 3991680, 2217600, 1663200, 1425600, 1330560, 43545600, 23950080, 17740800, 14968800, 13685760, 6652800, 518918400

Offset: 1

Alford Arnold, Jan 11 2001

These values also appear in a wider context when counting elements of finite groups by cycle structure. For example, the alternating group on four symbols has 12 elements; eight associated with the partition 3+1, three associated with 2+2 and the identity associated with 1+1+1+1. The cross-referenced sequences are all associated with similar numeric partitions and "M2" weights.

			Triangle begins:
  0;
  1;
  3;
  8, 3;
  30, 20;
  144, 90, 40;
  840, 504, 420;
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000012, A000035, A000027, A004526, A022003, A008619, A000217, A007997, A001399, A011765 A008620, A027656, A002620, A000292, A008627.

From Sean A. Irvine, Sep 05 2022: (Start)
T(1,1) = 0.
T(n,k) = n! / (k * (n-k)) for 1 <= k < n/2.
T(2n,n) = (2*n)! / (2*n^2).
(End)

More terms from Sean A. Irvine, Sep 05 2022

cp's OEIS Frontend

A118923 Triangle T(n,k) built by placing T(n,0)=A000012(n) in the left edge, T(n,n)=A079978(n) on the right edge and filling the body with the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A029047 Expansion of 1/((1-x)*(1-x^3)*(1-x^6)*(1-x^10)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A129920 Expansion of -1/(1 - x + 3*x^2 - 2*x^3 + x^4 - 2*x^5 + x^6).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A141828 a(n) = (n^4*a(n-1)-1)/(n-1) for n >= 2, with a(0) = 1, a(1) = 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A331943 a(n) = n^2 + 1 - ceiling((n + 2)/3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A348388 Irregular triangle read by rows: T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A029047 Expansion of 1/((1-x)(1-x^3)(1-x^6)*(1-x^10)).

A129920 Expansion of -1/(1 - x + 3x^2 - 2x^3 + x^4 - 2*x^5 + x^6).