A005586 -id:A005586 - cp's OEIS frontend

A129936 a(n) = (n-2)(n+3)(n+2)/6.

-2, -2, 0, 5, 14, 28, 48, 75, 110, 154, 208, 273, 350, 440, 544, 663, 798, 950, 1120, 1309, 1518, 1748, 2000, 2275, 2574, 2898, 3248, 3625, 4030, 4464, 4928, 5423, 5950, 6510, 7104, 7733, 8398, 9100, 9840, 10619, 11438, 12298, 13200, 14145, 15134, 16168, 17248

Offset: 0

Roger L. Bagula, Jun 09 2007

Essentially the same as A005586.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A002378, A005586, A023444, A034856, A214292.

seq(sum(i*(k-i+1), i=1..k+2), k=0..99); # Wesley Ivan Hurt, Sep 21 2013

f[n_] = Binomial[n + 3, 3] - (n + 3)*(n + 2)/2; Table[f[n], {n, 0, 30}]
LinearRecurrence[{4,-6,4,-1},{-2,-2,0,5},50] (* Harvey P. Dale, Jul 03 2020 *)

a(n)=(n-2)*(n+3)*(n+2)/6 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = binomial(n+3,3) - (n + 3)*(n + 2)/2.
a(n) = A214292(n+2,2). - Reinhard Zumkeller, Jul 12 2012
G.f.: (x^3-4*x^2+6*x-2)/(x-1)^4. - Colin Barker, Sep 05 2012
From Wesley Ivan Hurt, Sep 21 2013: (Start)
a(n) = Sum_{i=1..n+2} i*(n-i+1).
a(n+2) = A000292(n+1) + A034856(n), n>0. (End)
From Amiram Eldar, Sep 26 2022: (Start)
Sum_{n>=3} 1/a(n) = 77/200.
Sum_{n>=3} (-1)^(n+1)/a(n) = 363/200 - 12*log(2)/5. (End)
From Elmo R. Oliveira, Aug 04 2025: (Start)
E.g.f.: exp(x)*(x^3 + 6*x^2 - 12)/6.
a(n) = A023444(n)*A002378(n+2)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

More terms from Wesley Ivan Hurt, Sep 21 2013

A192174 Triangle T(n,k) of the coefficients [x^(n-k)] of the polynomial p(0,x)=-1, p(1,x)=x and p(n,x) = x*p(n-1,x) - p(n-2,x) in row n, column k.

Original entry on oeis.org

-1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, -1, 1, 0, -2, 0, -1, 0, 1, 0, -3, 0, 0, 0, 1, 1, 0, -4, 0, 2, 0, 2, 0, 1, 0, -5, 0, 5, 0, 2, 0, -1, 1, 0, -6, 0, 9, 0, 0, 0, -3, 0, 1, 0, -7, 0, 14, 0, -5, 0, -5, 0, 1, 1, 0, -8, 0, 20, 0, -14, 0, -5, 0, 4, 0

Offset: 0

Paul Curtz, Jun 24 2011

Consider the Catalan triangle A009766 antisymmetrically extended by a mirror along the diagonal (see also A176239):
0, -1, -1, -1, -1,  -1,  -1,   -1,
1,  0, -1, -2, -3,  -4,  -5,   -6,
1,  1,  0, -2, -5,  -9, -14,  -20,
1,  2,  2,  0, -5, -14, -28,  -48,
1,  3,  5,  5,  0, -14, -42,  -90,
1,  4,  9, 14, 14,   0, -42, -132,
1,  5, 14, 28, 42,  42,   0, -132,
1,  6, 20, 48, 90, 132, 132,    0.
The rows in this array are essentially the columns of T(n,k).

			Triangle begins
  -1;      # -1
   1,  0;      # x
   1,  0,  1;      # x^2+1
   1,  0,  0,  0;      # x^3
   1,  0, -1,  0, -1;      # x^4-x^2-1
   1,  0, -2,  0, -1,  0;
   1,  0, -3,  0,  0,  0,  1;
   1,  0, -4,  0,  2,  0,  2,  0;
   1,  0, -5,  0,  5,  0,  2,  0, -1;
   1,  0, -6,  0,  9,  0,  0,  0, -3,  0;
   1,  0, -7,  0, 14,  0, -5,  0, -5,  0,  1;
   1,  0, -8,  0, 20,  0,-14,  0, -5,  0,  4,  0;
   1,  0, -9,  0, 27,  0,-28,  0,  0,  0,  9,  0, -1;

Cf. A023443, A080956 and A000096, A129936 and A005586, A005587.

Cf. A194084. - Paul Curtz, Aug 16 2011

p:= proc(n,x) option remember: if n=0 then -1 elif n=1 then x elif n>=2 then x*procname(n-1,x)-procname(n-2,x) fi: end: A192174 := proc(n,k): coeff(p(n,x),x,n-k): end: seq(seq(A192174(n,k),k=0..n), n=0..11); # Johannes W. Meijer, Aug 21 2011

Sum_{k=0..n} T(n,k) = A057079(n-1).

Apparently T(3s,2s-2) = (-1)^(s+1)*A000245(s), s >= 1.

A205574 Triangle T(n,k), 0<=k<=n, given by (0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 15, 14, 9, 4, 1, 0, 52, 44, 28, 14, 5, 1, 0, 203, 154, 93, 48, 20, 6, 1, 0, 877, 595, 333, 169, 75, 27, 7, 1, 0, 4140, 2518, 1289, 624, 280, 110, 35, 8, 1, 0, 21147, 11591, 5394, 2442, 1071, 435, 154, 44, 9, 1

Offset: 0

Philippe Deléham, Jan 29 2012

Bell convolution triangle ; g.f. for column k : (x*B(x))^k with  B(x) g.f. for A000110 (Bell numbers).
Riordan array (1, x*B(x)), when B(x) the g.f. of A000110.
Row sums are in A137551.

			Triangle begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,  1;
  0,   5,   5,  3,  1;
  0,  15,  14,  9,  4,  1;
  0,  52,  44, 28, 14,  5, 1;
  0, 203, 154, 93, 48, 20, 6, 1;
  ...

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

Cf. Columns : A000007, A000110, A014322, A014323, A014325 ; Diagonals : A000012, A001477, A000096, A005586.
Another version: A292870.
T(2n,n) gives: A292871.

# Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-bell(n-1)); # Peter Luschny, Oct 19 2022

Sum_{k=0..n} T(n,k) = A137551(n), n>0.

A024191 [ (3rd elementary symmetric function of 3,4,...,n+4)/(3+4+...+n+4) ].

Original entry on oeis.org

5, 19, 47, 95, 170, 280, 434, 642, 915, 1265, 1705, 2249, 2912, 3710, 4660, 5780, 7089, 8607, 10355, 12355, 14630, 17204, 20102, 23350, 26975, 31005, 35469, 40397, 45820, 51770, 58280, 65384, 73117, 81515, 90615, 100455, 111074, 122512, 134810, 148010

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A115127.

Partial sums of A005586.

Table[n(n+1)(n^2+13n+46)/24,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{5,19,47,95,170},40] (* Harvey P. Dale, Apr 28 2014 *)
CoefficientList[Series[(5 - 6 x + 2 x^2)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 28 2014 *)

a(n) = n*(n+1)*(n^2+13*n+46)/24 \\ Charles R Greathouse IV, Oct 21 2022

a(n)=A115127(n+2, 3), n>=2.
a(n) = n*(n+1)*(n^2+13n+46)/24 =a(n-1)+A005586(n). - Henry Bottomley, Oct 25 2001
G.f.: x*(5-6*x+2*x^2)/(1-x)^5.
a(n) = floor(A024184(n)/A055998(n+2)). -  R. J. Mathar, Sep 15 2009
a(1)=5, a(2)=19, a(3)=47, a(4)=95, a(5)=170, a(n)=5*a(n-1)- 10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Apr 28 2014

A176239 Shifted signed Catalan triangle T(n,k) = (-1)^(n+k+1)A009766(n,k-n+1) read by rows.

Original entry on oeis.org

0, -1, 1, -1, 0, 2, 0, 1, -2, 2, 0, -5, 0, 0, 1, -3, 5, -5, 0, 14, 0, 0, 0, 1, -4, 9, -14, 14, 0, -42, 0, 0, 0, 0, 1, -5, 14, -28, 42, -42, 0, 132, 0, 0, 0, 0, 0, 1, -6, 20, -48, 90, -132, 132, 0, -429, 0, 0, 0, 0, 0, 0, 1, -7, 27, -75, 165, -297, 429, -429, 0, 1430

Offset: 0

Paul Curtz, Apr 12 2010

			The triangle starts in row n=0 with columns 0 <= k < 2*(n+1) as:
0,-1;                          (-1)^k*k  A001477
1,-1,.0,.2;                      (-1)^(k+1)*(k+1)*(k-2)/2  A080956, A000096
0,.1,-2,.2,.0,-5;                 (-1)^n*k*(k+1)*(k-4)/6 A129936, A005586
0,.0,.1,-3,.5,-5,..0,.14;             (-1)^k*k*(k+1)*(k-1)*(k-6)/24, A005587
0,.0,.0,.1,-4,.9,-14,.14,.0,-42;           A005557, A034807
0,.0,.0,.0,.1,-5,.14,-28,42,-42,0,132;

Cf A140344, A033184.

A009766 := proc(n,k) if k<0 or k >n then 0; else binomial(n+k,n)*(n-k+1)/(n+1) ; end if; end proc:
A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A176239 := proc(n,k) if k <= 2*n-1 then (-1)^(n+k+1)*A009766(n,k-n+1) elif k = 2*n then 0; elif k < 2*(n+1) then (-1)^(n+1)*A000108(n+1); else 0; end if; end proc: # R. J. Mathar, Dec 03 2010

T(n,k) = T(n+1,k)+T(n+1,k+1), k <= 2n+1.
T(n,2n) = 0.
T(n,2n+1) = (-1)^(n+1)*A000108(n+1).
T(n,k) =  (-1)^(n+k+1)*A009766(n,k-n+1), k < 2n.

A212393 Expansion of (1-4x+7x^2-5x^3+4x^4-6x^5+21x^6+18x^7-5x^8)/(1-x)^5.

Original entry on oeis.org

1, 1, 2, 5, 14, 30, 72, 195, 485, 1059, 2065, 3682, 6120, 9620, 14454, 20925, 29367, 40145, 53655, 70324, 90610, 115002, 144020, 178215, 218169, 264495, 317837, 378870, 448300, 526864, 615330, 714497, 825195, 948285, 1084659, 1235240, 1400982, 1582870, 1781920

Offset: 0

Bruno Berselli, May 14 2012

In the paper of Kitaev, Remmel and Tiefenbruck (see the Links section), Q_(132)^(0,k,0,0)(x,0) represents a generating function depending on k and x.
For successive values of k we have:
k=1, the g.f. of A000012: 1/(1-x);
k=2, the g.f. of A028310: (1-x+x^2)/(1-x)^2;
k=3, the g.f. (1-2*x+2*x^2+x^3-x^4)/(1-x)^3, whose coefficients (except the first two) are given by A000096 (for n>0);
k=4, the g.f. (1-3*x+4*x^2-x^3+3*x^4-5*x^5+2*x^6)/(1-x)^4, whose coefficients (except the first three) are given by A005586 (for n>0).
This sequence corresponds to the case k=5.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (page 21, Theorem 10).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

m:=39; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5));

CoefficientList[Series[(1 - 4 x + 7 x^2 - 5 x^3 + 4 x^4 - 6 x^5 + 21 x^6 + 18 x^7 - 5 x^8)/(1 - x)^5, {x, 0, 38}], x]

Vec((1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5+O(x^39))

G.f.: (1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>8, a(0)=a(1)=1, a(2)=2, a(3)=5, a(4)=14, a(5)=30, a(6)=72, a(7)=195, a(8)=485.
a(n) = (n-3)*(31*n^3-369*n^2+1454*n-1560)/24 for n>3, a(0)=a(1)=1, a(2)=2, a(3)=5.
G.f.: 1+x+2*x^2+5*x^3 + 14*x^4*G(0), where G(k)= 1 + x*(k+1)*(124*k^3+192*k^2+89*k+180)/( (2*k+1)*(62*k^3+3*k^2-5*k+84) - x*(2*k+1)*(62*k^3+3*k^2-5*k+84)*(2*k+3)*(62*k^3+189*k^2+187*k+144)/(x*(2*k+3)*(62*k^3+189*k^2+187*k+144) + (k+1)*(124*k^3+192*k^2+89*k+180)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013

A254749 1-gonal pyramidal numbers.

Original entry on oeis.org

1, 2, 2, 0, -5, -14, -28, -48, -75, -110, -154, -208, -273, -350, -440, -544, -663, -798, -950, -1120, -1309, -1518, -1748, -2000, -2275, -2574, -2898, -3248, -3625, -4030, -4464, -4928, -5423, -5950, -6510, -7104, -7733, -8398, -9100, -9840, -10619, -11438

Offset: 1

Colin Barker, Feb 07 2015

Not strictly pyramidal numbers, but the result of using the Wikipedia formula with r = 1.

Essentially the same as A129936 and A005586.

			G.f. = x + 2*x^2 + 2*x^3 - 5*x^5 - 14*x^6 - 28*x^7 - 48*x^8 - 75*x^9 + ...

Colin Barker, Table of n, a(n) for n = 1..1000
Wikipedia, Pyramidal number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005581, A005586, A129936.

[(n*(4+3*n-n^2))/6: n in [1..60]]; // G. C. Greubel, Aug 03 2018

Table[(n*(4+3*n-n^2))/6, {n,1,60}] (* or *) LinearRecurrence[{4,-6,4,-1}, {1, 2, 2, 0}, 60] (* G. C. Greubel, Aug 03 2018 *)

ppg(r, n) = (3*n^2+n^3*(r-2)-n*(r-5))/6
vector(100, n, ppg(1, n))

a(n) = n*(4 + 3*n - n^2)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(1 - 2*x)/(x-1)^4.
a(n) = A005581(-n) = -A005586(n-4) = -A129936(n-2) for all n in Z. - Michael Somos, Jul 28 2015
E.g.f.: -exp(x)*x*(x^2 - 6)/6. - Elmo R. Oliveira, Aug 04 2025

A115126 First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).

Original entry on oeis.org

1, 2, 2, 3, 5, 5, 4, 9, 14, 14, 5, 14, 28, 42, 42, 6, 20, 48, 90, 132, 132, 7, 27, 75, 165, 297, 429, 429, 8, 35, 110, 275, 572, 1001, 1430, 1430, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796, 16796, 11, 65, 273, 910

Offset: 1

Wolfdieter Lang, Jan 13 2006

First (k=0) column removed from Catalan triangle A009766(n,k).
In the Derrida et al. 1992 reference this triangle, called here X(alpha=1,beta=1;k=1,n,m), n >= m >= 1, is called there X_{N=n}(K=1,p=m) with alpha=1 and beta=1.
The column sequences give A000027 (natural numbers), A000096, A005586, A005587, A005557, A064059, A064061 for m=1..7. The numerator polynomials for the o.g.f. of column m is found in A062991 and the denominator is (1-x)^(m+1).
The diagonal sequences are convolutions of the Catalan numbers A000108, starting with the main diagonal.

			[1];[2,2];[3,5,5];[4,9,14,14];...
a(4,2) = 9 = binomial(6,2)*3/5.

B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.

W. Lang: First 10 rows.

Row sums give A001453(n+1)=A000108(n+1)-1 (Catalan -1).

a(n, m)= binomial(n+m, n)*(n-m+1)/(n+1), n>=m>=1; a(n, m)=0 if n

A176270 Triangle T(n,m) = 1 + m*(m-n) read by rows, 0 <= m <= n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -3, -2, 1, 1, -3, -5, -5, -3, 1, 1, -4, -7, -8, -7, -4, 1, 1, -5, -9, -11, -11, -9, -5, 1, 1, -6, -11, -14, -15, -14, -11, -6, 1, 1, -7, -13, -17, -19, -19, -17, -13, -7, 1, 1, -8, -15, -20, -23, -24, -23, -20, -15, -8, 1

Offset: 0

Roger L. Bagula, Apr 13 2010

For GCD(-1 - m,-1 - n + m) = 1, smallest number that cannot be written as a*(-1 - m) + b*(-1 - n + m) with a and b in the nonnegative integers. - Thomas Anton, May 22 2019

			Triangle begins
  1;
  1,   1;
  1,   0,   1;
  1,  -1,  -1,   1;
  1,  -2,  -3,  -2,   1;
  1,  -3,  -5,  -5,  -3,   1;
  1,  -4,  -7,  -8,  -7,  -4,   1;
  1,  -5,  -9, -11, -11,  -9,  -5,   1;
  1,  -6, -11, -14, -15, -14, -11,  -6,   1;
  1,  -7, -13, -17, -19, -19, -17, -13,  -7,   1;
  1,  -8, -15, -20, -23, -24, -23, -20, -15,  -8,   1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A005586 (row sums), A077028.

Flat(List([0..12], n-> List([0..n], k-> k*(k-n)+1 ))); # G. C. Greubel, May 30 2019

[[k*(k-n)+1: k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 30 2019

A176270 := proc(n,m)
        1+m*(m-n) ;
end proc: # R. J. Mathar, May 03 2013

Table[k*(k-n)+1, {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 30 2019 *)

{T(n,k) = k*(k-n)+1}; \\ G. C. Greubel, May 30 2019

[[k*(k-n)+1 for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 30 2019

T(n,m) = binomial(n-m+1,2) + binomial(m+1,2) - binomial(n+1,2) + 1 = m^2 - n*m + 1.
T(n,m) = T(n,n-m).
T(n,m) = 2 - A077028(n,m) for 0 <= m <= n. - Werner Schulte, Nov 10 2020

Edited by R. J. Mathar, May 03 2013

Previous Showing 11-19 of 19 results.

cp's OEIS Frontend

A129936 a(n) = (n-2)*(n+3)*(n+2)/6.

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A192174 Triangle T(n,k) of the coefficients [x^(n-k)] of the polynomial p(0,x)=-1, p(1,x)=x and p(n,x) = x*p(n-1,x) - p(n-2,x) in row n, column k.

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A205574 Triangle T(n,k), 0<=k<=n, given by (0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A024191 [ (3rd elementary symmetric function of 3,4,...,n+4)/(3+4+...+n+4) ].

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A176239 Shifted signed Catalan triangle T(n,k) = (-1)^*(n+k+1)*A009766(n,k-n+1) read by rows.

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A212393 Expansion of (1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5.

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A254749 1-gonal pyramidal numbers.

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A115126 First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).

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A129936 a(n) = (n-2)(n+3)(n+2)/6.

A176239 Shifted signed Catalan triangle T(n,k) = (-1)^(n+k+1)A009766(n,k-n+1) read by rows.

A212393 Expansion of (1-4x+7x^2-5x^3+4x^4-6x^5+21x^6+18x^7-5x^8)/(1-x)^5.