A144328 -id:A144328 - cp's OEIS frontend

A064999 Partial sums of sequence (essentially A002378): 1, 2, 6, 12, 20, 30, 42, 56, 72, 90, ...

1, 3, 9, 21, 41, 71, 113, 169, 241, 331, 441, 573, 729, 911, 1121, 1361, 1633, 1939, 2281, 2661, 3081, 3543, 4049, 4601, 5201, 5851, 6553, 7309, 8121, 8991, 9921, 10913, 11969, 13091, 14281, 15541, 16873, 18279, 19761, 21321, 22961, 24683

Offset: 0

Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Oct 31 2001

Equals triangle A144328 * [1, 2, 3, ...]. - Gary W. Adamson, Sep 18 2008

a(n) is the number of parking functions of size n+1 avoiding the patterns 123 and 312. - Lara Pudwell, Apr 10 2023

Harry J. Smith, Table of n, a(n) for n = 0..1000
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Franck Ramaharo, Enumerating the states of the twist knot, arXiv:1712.06543 [math.CO], 2017.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002378, A007290.

Cf. A144328. - Gary W. Adamson, Sep 18 2008

[(n^3+3*n^2+2*n+3)/3: n in [0..50]]; // Vincenzo Librandi, Feb 28 2016

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^2-n od: seq(a[n], n=0..42); # Zerinvary Lajos, Jun 05 2008

Table[(x^3 - x + 3)/3, {x, 1, 100}] (* Artur Jasinski, Feb 14 2007 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 3, 9, 21}, 50] (* Vincenzo Librandi, Feb 28 2016 *)

{ for (n=0, 1000, if (n, a+=n*(n + 1), a=1); write("b064999.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 03 2009

a(n) = (n^3+3*n^2+2*n+3)/3; \\ Altug Alkan, May 16 2018

a(n) = A007290(n+2) + 1 = (n^3 + 3*n^2 + 2*n + 3)/3.
a(0) = 1, a(n) = n*(n+1) + a(n-1) for n > 1. - Gerald McGarvey, Sep 26 2004
O.g.f.: (1 - x + 3x^2 - x^3)/(1 - x)^4.

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Nov 12 2001

A176484 Triangle t(n,m) read by rows: t(n,n)= 1, t(n,m) = (-1)^(n+m)*(m+1), 0<=m

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -1, 2, -3, 1, 1, -2, 3, -4, 1, -1, 2, -3, 4, -5, 1, 1, -2, 3, -4, 5, -6, 1, -1, 2, -3, 4, -5, 6, -7, 1, 1, -2, 3, -4, 5, -6, 7, -8, 1, -1, 2, -3, 4, -5, 6, -7, 8, -9, 1, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 1

Offset: 0

Roger L. Bagula, Apr 18 2010

Row sums are 1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -4,..

Obtained from A144328 by deleting the first column, adding a diagonal of +1's, and multiplication with (-1)^(n-m).

			1;
-1, 1;
1, -2, 1;
-1, 2, -3, 1;
1, -2, 3, -4, 1;
-1, 2, -3, 4, -5, 1;
1, -2, 3, -4, 5, -6, 1;
-1, 2, -3, 4, -5, 6, -7, 1;
1, -2, 3, -4, 5, -6, 7, -8, 1;
-1, 2, -3, 4, -5, 6, -7, 8, -9, 1;
1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 1;

p[x, 0] = 1;
p[x_, n_] := p[x, n] = x^n - Sum[(-1)^(i + Mod[n, 2])*i*x^(i - 1), {i, 1, n}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

t(n,m) = [x^m] (x^n - sum_{j=1..n} (-1)^(j+(n mod 2)) *j*x^(j-1) ) .

A178702 Coefficients of the sum 1+ x/(1-x) + x^2/(1-x^2) + x^3/ ( (1-x)(1-x^2)) + x^4/ ( (1-x)(1-x^3) ) + x^5/ ( (1-x)(1-x^4) ) + x^5 /((1-x^2)(1-x^3)) + x^6/ ( (1-x)(1-x^2)(1-x^3)) + ...

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 7, 10, 14, 20, 24, 32, 40, 54, 69, 86, 106, 135, 165, 206, 256, 311, 378, 460, 555, 670, 808, 970, 1156, 1380, 1638, 1938, 2296, 2706, 3188, 3752, 4390, 5136, 6000, 6990, 8128, 9444, 10944, 12672, 14659, 16904, 19476, 22420, 25753, 29550, 33873, 38759, 44306

Offset: 0

David S. Newman, Dec 27 2010

nonn

For a given positive integer, n, let S_n be the set of partitions of n into distinct parts where the number of parts is maximal for that n.  For example, for n=6, the set S_6 consists of just one such partition S_6={1,2,3}.  Similarly, for n=7, S_7={1,2,4},  But for n=8, S_8 will contain two partitions S_8= { {1,2,5}, {1,3,4} }.  Now form the sum whose general term is x^n divided by the product (1-x^(p_1))...(1-x^(p_i)) where the p's come from the partitions in S_n.  The sequence is the sequence of coefficients of this sum.
This sequence is an upper bound for A098859.
The initial values of the sequence were verified by several members of the Sequence Fans mailing list.
 |S_n| = A144328(n) (if the offset of the latter is changed to 0). - Benoit Jubin, Dec 13 2010.

Cf. A098859, A144328.

Mathematica
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More terms from Alois P. Heinz, Jun 25 2011

cp's OEIS Frontend

A064999 Partial sums of sequence (essentially A002378): 1, 2, 6, 12, 20, 30, 42, 56, 72, 90, ...

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A176484 Triangle t(n,m) read by rows: t(n,n)= 1, t(n,m) = (-1)^(n+m)*(m+1), 0<=m

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A178702 Coefficients of the sum 1+ x/(1-x) + x^2/(1-x^2) + x^3/ ( (1-x)(1-x^2)) + x^4/ ( (1-x)(1-x^3) ) + x^5/ ( (1-x)(1-x^4) ) + x^5 /((1-x^2)(1-x^3)) + x^6/ ( (1-x)(1-x^2)(1-x^3)) + ...

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