A154990 -id:A154990 - cp's OEIS frontend

A080956 a(n) = (n+1)*(2-n)/2.

1, 1, 0, -2, -5, -9, -14, -20, -27, -35, -44, -54, -65, -77, -90, -104, -119, -135, -152, -170, -189, -209, -230, -252, -275, -299, -324, -350, -377, -405, -434, -464, -495, -527, -560, -594, -629, -665, -702, -740, -779, -819, -860, -902, -945, -989, -1034, -1080, -1127, -1175, -1224, -1274, -1325, -1377

Offset: 0

Paul Barry, Mar 01 2003

Coefficient of x in the polynomial C(n,0)+C(n+1,1)x+C(n+2,2)x(x-1)/2.
Equals A154990 * [1,2,3,...]. - Gary W. Adamson & Mats Granvik, Jan 19 2009
a(n) is essentially the case 1 of the polygonal numbers. The polygonal numbers are defined as P_k(n) = Sum_{i=1..n} ((k-2)*i-(k-3)). Thus P_1(n) = n*(3-n)/2 and a(n) = P_1(n+1). See A005563 for the case k=0. - Peter Luschny, Jul 08 2011
This is the case k=-1 of the formula (k*m*(m+1)-(-1)^k+1)/2. See similar sequences listed in A262221. - Bruno Berselli, Sep 17 2015

			a(5) = 6-(1+2+3+4+5). - _Stanislav Sykora_, Feb 19 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000027, A000096, A000217, A154990, A214292, A262221.

[(n+1)*(2-n)/2: n in [0..80]]; // Vincenzo Librandi, Jul 08 2011

G(x):=exp(x)*(x-x^2/2): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..54 ); # Zerinvary Lajos, Apr 05 2009

FoldList[#1 - #2 &, 1, Range[0, 44]] (* Arkadiusz Wesolowski, May 26 2013 *)
LinearRecurrence[{3,-3,1},{1,1,0},60] (* Harvey P. Dale, Nov 29 2019 *)

PARI
```
a(n)=(n+1)*(2-n)/2;
```

def A080956(n): return (2-n)*(n+1)//2 # G. C. Greubel, May 08 2025

a(n) = 2*(C(n+1, 1)-C(n+2, 2)) = (n+1)*(2-n)/2.
G.f.: (1-2*x)/(1-x)^3. - R. J. Mathar, Jun 11 2009
If we define f(n,i,a) = Sum_{k=0..n-i} (binomial(n,k)*stirling1(n-k,i)*Product_{j=0..k-1} (-a-j)), then a(n) = f(n,n-1,2), for n>=3. - Milan Janjic, Dec 20 2008
E.g.f.: exp(x)*(1-x^2/2). - Zerinvary Lajos, Apr 05 2009, R. J. Mathar, Jun 11 2009
a(n) = - A214292(n,1) for n > 0. - Reinhard Zumkeller, Jul 12 2012
Recurrence: a(0)=1, a(n+1) = a(n) - n. Also a(n)=(n+1)-Sum[k=1..n](k). Also a(n) = A000027(n+1) - A000217(n). Also, for n>1, a(n) = - A000096(n-2). - Stanislav Sykora, Feb 19 2014
Sum_{n>=3} 1/a(n) = -11/9. - Amiram Eldar, Sep 26 2022

Lajos e.g.f. adapted to offset zero by R. J. Mathar, Jun 11 2009

A092582 Triangle read by rows: T(n,k) is the number of permutations p of [n] having length of first run equal to k.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 12, 8, 3, 1, 60, 40, 15, 4, 1, 360, 240, 90, 24, 5, 1, 2520, 1680, 630, 168, 35, 6, 1, 20160, 13440, 5040, 1344, 280, 48, 7, 1, 181440, 120960, 45360, 12096, 2520, 432, 63, 8, 1, 1814400, 1209600, 453600, 120960, 25200, 4320, 630, 80, 9, 1

Offset: 1

Emeric Deutsch and Warren P. Johnson (wjohnson(AT)bates.edu), Apr 10 2004

Row sums are the factorial numbers (A000142). First column is A001710.
T(n,k) = number of permutations of [n] in which 1,2,...,k is a subsequence but 1,2,...,k,k+1 is not. Example: T(4,2)=8 because 1324, 1342, 1432, 4132, 3124, 3142, 3412 and 4312, are the only permutations of [4] in which 12 is a subsequence but 123 is not. - Emeric Deutsch, Nov 12 2004
T(n,k) is the number of deco polyominoes of height n with k cells in the last column. (A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column). - Emeric Deutsch, Jan 06 2005
T(n,k) is the number of permutations p of [n] for which the smallest i such that p(i)Emeric Deutsch, Feb 23 2008
Adding columns 2,4,6,... one obtains the derangement numbers 0,1,2,9,44,... (A000166). See the Bona reference (p. 118, Exercises 41,42). - Emeric Deutsch, Feb 23 2008
Matrix inverse of A128227*A154990. - Mats Granvik, Feb 08 2009
Differences in the columns of A173333 which counts the n-permutations with an initial ascending run of length at least k. - Geoffrey Critzer, Jun 18 2017
The triangle with each row reversed is A130477. - Michael Somos, Jun 25 2017

			T(4,3) = 3 because 1243, 1342 and 2341 are the only permutations of [4] having length of first run equal to 3.
     1;
     1,    1;
     3,    2,   1;
    12,    8,   3,   1;
    60,   40,  15,   4,  1;
   360,  240,  90,  24,  5,  1;
  2520, 1680, 630, 168, 35,  6,  1;
  ...

M. Bona, Combinatorics of Permutations, Chapman&Hall/CRC, Boca Raton, Florida, 2004.

Alois P. Heinz, Rows n = 1..141, flattened
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Colin Defant and James Propp, Quantifying Noninvertibility in Discrete Dynamical Systems, arXiv:2002.07144 [math.CO], 2020.
Emeric Deutsch and W. P. Johnson, Create your own permutation statistics, Math. Mag., 77, 130-134, 2004.

Cf. A002260. - Gary W. Adamson, Feb 22 2012
Cf. A000166, A002627, A055596, A092580, A092956, A130477.
Cf. A000027, A001710, A001715, A001720, A001725, A005563.
Cf. A000522.

Flat(List([1..11],n->Concatenation([1],List([1..n-1],k->Factorial(n)*k/Factorial(k+1))))); # Muniru A Asiru, Jun 10 2018

A092582:= func< n,k | k eq n select 1 else k*Factorial(n)/Factorial(k+1) >;
[A092582(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Sep 06 2022

Drop[Drop[Abs[Map[Select[#, # < 0 &] &, Map[Differences, nn = 10; Range[0, nn]! CoefficientList[Series[(Exp[y x] - 1)/(1 - x), {x, 0, nn}], {x, y}]]]], 1], -1] // Grid (* Geoffrey Critzer, Jun 18 2017 *)

{T(n, k) = if( n<1 || k>n, 0, k==n, 1, n! * k /(k+1)!)}; /* Michael Somos, Jun 25 2017 */

def A092582(n,k): return 1 if (k==n) else k*factorial(n)/factorial(k+1)
flatten([[A092582(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Sep 06 2022

T(n, k) = n!*k/(k+1)! for k

Inverse of:

-1, 1;

-1, -2, 1;

-1, -2, -3, 1;

-1, -2, -3, -4, 1;

... where A002260 = (1; 1,2; 1,2,3; ...). - Gary W. Adamson, Feb 22 2012

T(2n,n) = A092956(n-1) for n>0. - Alois P. Heinz, Jun 19 2017

From Alois P. Heinz, Dec 17 2021: (Start)

Sum_{k=1..n} k * T(n,k) = A002627(n).

|Sum_{k=1..n} (-1)^k * T(n,k)| = A055596(n) for n>=1. (End)

From G. C. Greubel, Sep 06 2022: (Start)

T(n, 1) = A001710(n).

T(n, 2) = 2*A001715(n) + [n=2]/3, n >= 2.

T(n, 3) = 3*A001720(n) + [n=3]/4, n >= 3.

T(n, 4) = 4*A001725(n) + [n=4]/5, n >= 4.

T(n, n-1) = A000027(n-1).

T(n, n-2) = A005563(n-1), n >= 3. (End)

Sum_{k=0..n} (k+1) * T(n,k) = A000522(n). - Alois P. Heinz, Apr 28 2023

A145839 Number of 3-compositions of n.

Original entry on oeis.org

1, 3, 15, 73, 354, 1716, 8318, 40320, 195444, 947380, 4592256, 22260144, 107902088, 523036176, 2535324816, 12289536016, 59571339552, 288761470848, 1399719859808, 6784893012864, 32888561860032, 159421452802624, 772767131681280, 3745851196992000

Offset: 0

Simone Rinaldi (rinaldi(AT)unisi.it), Oct 21 2008

A 3-composition of n is a matrix with three rows, such that each column has at least one nonzero element and whose elements sum up to n.
Matrix inverse of (A000217(A004736)*A154990). - Mats Granvik, Jan 19 2009
(1 +3*x +15*x^2 +73*x^3 + ...) = 1/(1 -3*x -6*x^2 -10*x^3 -15*x^4 - ...). - Gary W. Adamson, Jul 27 2009
For n>1, a(n) is the number of generalized compositions of n-1 when there are i^2/2 +3i/2 +1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

G. Louchard, Matrix compositions: a probabilistic approach, Proceedings of GASCom and Bijective Combinatorics 2008, Bibbiena, Italy, pp. 159-170.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 8.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Emanuele Munarini, Maddalena Poneti, and Simone Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8.
Index entries for linear recurrences with constant coefficients, signature (6,-6,2).

Cf. A003480 (2-compositions), A145840 (4-compositions), A145841 (5-compositions).

Column k=3 of A261780.

I:=[3,15,73]; [1] cat [n le 3 select I[n] else 6*Self(n-1) - 6*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Mar 07 2021

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*binomial(j+2, 2), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 01 2015

Table[Sum[Binomial[n+3*k-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* Vaclav Kotesovec, Dec 31 2013 *)
a[n_]:= a[n]= If[n==0, 1, Sum[Binomial[n-j+2, 2]*a[j], {j,0,n-1}]]; Table[a[n], {n, 0, 20}] (* G. C. Greubel, Mar 07 2021 *)

@CachedFunction
def a(n):
    if n==0: return 1
    else: return sum( binomial(n-j+2,2)*a(j) for j in (0..n-1))
[a(n) for n in (0..25)] # G. C. Greubel, Mar 07 2021

a(n+3) = 6*a(n+2) - 6*a(n+1) + 2*a(n).
G.f.: (1-x)^3/(2*(1-x)^3 - 1).
a(n) = Sum_{k>=0} C(n+3*k-1,n) / 2^(k+1). - Vaclav Kotesovec, Dec 31 2013
a(n) = Sum_{j=0..n-1} binomial(n-j+2, 2)*a(j) with a(0) = 1. - G. C. Greubel, Mar 07 2021

Offset corrected by Alois P. Heinz, Aug 31 2015

A155038 Triangle read by rows: T(n,k) is the number of compositions of n with first part k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 4, 2, 1, 1, 16, 8, 4, 2, 1, 1, 32, 16, 8, 4, 2, 1, 1, 64, 32, 16, 8, 4, 2, 1, 1, 128, 64, 32, 16, 8, 4, 2, 1, 1, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1, 2048, 1024, 512

Offset: 1

Mats Granvik, Jan 19 2009

Previous name was: Matrix inverse of A154990.
Apart from first term essentially the same as A057728.
A011782 appears in the columns.
Riordan array ((1-x)/(1-2x), x). - Philippe Deléham, Jan 24 2010
Indexing the triangle from n=0 and k=0, T(n,k) is the number of binary words of length n that begin with a run of exactly k 0's. O.g.f.: 1/((1-y*x)*(1-x/(1-x))). - Geoffrey Critzer, Feb 15 2012

			T(5,2) = 4 because the compositions of 5 with first part 2 are: [2,3], [2,2,1], [2,1,2], and [2,1,1,1]. - _Emeric Deutsch_, Jan 12 2018
Table begins:
   1,
   1,  1,
   2,  1,  1,
   4,  2,  1,  1,
   8,  4,  2,  1,  1,
  16,  8,  4,  2,  1,  1,
  32, 16,  8,  4,  2,  1,  1,
  64, 32, 16,  8,  4,  2,  1,  1,
Production matrix begins:
  1, 1
  1, 0, 1
  1, 0, 0, 1
  1, 0, 0, 0, 1
  1, 0, 0, 0, 0, 1
  1, 0, 0, 0, 0, 0, 1
  1, 0, 0, 0, 0, 0, 0, 1
  1, 0, 0, 0, 0, 0, 0, 0, 1
  ... - _Philippe Deléham_, Oct 04 2014

Reinhard Zumkeller, Rows n = 1..100 of table, flattened
Jean-Luc Baril, Javier F. González, and José L. Ramírez, Last symbol distribution in pattern avoiding Catalan words, Univ. Bourgogne (France, 2022).
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007.

Cf. A011782, A057728, A154990, A155033, A155039.

a155038 n k = a155038_tabl !! (n-1) !! (k-1)
a155038_row n = a155038_tabl !! (n-1)
a155038_tabl = iterate
   (\row -> zipWith (+) (row ++ [0]) (init row ++ [0,1])) [1]
-- Reinhard Zumkeller, Aug 08 2013

T := proc(n, k) if k = n then 1 elif k < n then 2^(n-k-1) else 0 end if end proc: for n to 13 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jan 12 2018
G:= (1-2*x+t*x^2)/((1-2*x)*(1-t*x)): Gser := simplify(series(G, x = 0, 15)): for n to 14 do P[n] := coeff(Gser, x, n) end do: for n to 14 do seq(coeff(P[n], t, j), j = 1 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jan 19 2018

nn = 15; a = 1/(1 - y x); f[list_] := Select[list, # > 0 &];Map[f, CoefficientList[Series[ a/(1 - x/(1 - x)), {x, 0, nn}], {x, y}]] // Flatten (* Geoffrey Critzer, Feb 15 2012 *)

T(j,k) = A011782(j-k), j>=1, k>=1. - Omar E. Pol, Feb 14 2013
T(n,k) = 2^{n-k-1} if kn. - Emeric Deutsch, Jan 12 2018
G.f.: G(t,x) = (1-2*x+t*x^2)/((1-2*x)*(1-t*x)). - Emeric Deutsch, Jan 19 2018

New name from Joerg Arndt, May 04 2014

A155031 Triangle T(n, k) = 0 if n==0 (mod k) otherwise -1 with T(n, n) = 1 and T(n, 0) = 0, read by rows.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 0, -1, 1, 0, -1, -1, -1, 1, 0, 0, 0, -1, -1, 1, 0, -1, -1, -1, -1, -1, 1, 0, 0, -1, 0, -1, -1, -1, 1, 0, -1, 0, -1, -1, -1, -1, -1, 1, 0, 0, -1, -1, 0, -1, -1, -1, -1, 1, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 0, 0, 0, 0, -1, 0, -1, -1, -1, -1, -1, 1, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1

Offset: 1

Mats Granvik, Jan 19 2009

			Table begins:
  1;
  0,  1;
  0, -1,  1;
  0,  0, -1,  1;
  0, -1, -1, -1,  1;
  0,  0,  0, -1, -1,  1;
  0, -1, -1, -1, -1, -1,  1;
  0,  0, -1,  0, -1, -1, -1,  1;
  0, -1,  0, -1, -1, -1, -1, -1, 1;

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Cf. A154990, A155029.

[k eq n select 1 else (k eq 1 or n mod k eq 0) select 0 else -1: k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 08 2021

T[n_, k_]:= If[k==n, 1, If[k==1 || Mod[n, k]==0, 0, -1]];
Table[T[n, k], {n, 12}, {k, n}] //Flatten (* G. C. Greubel, Mar 08 2021 *)

flatten([[1 if k==n else 0 if (k==1 or n%k==0) else -1 for k in [1..n]] for n in [1..12]]) # G. C. Greubel, Mar 08 2021

T(n, k) = A154990(n, k) * A155029(n, k).

T(n, k) = 0 if n==0 (mod k) otherwise -1 with T(n, n) = 1 and T(n, 0) = 0.

A174557 Triangle T(n, k) = -floor(n/k) with T(n, n) = 1, read by rows.

Original entry on oeis.org

1, -2, 1, -3, -1, 1, -4, -2, -1, 1, -5, -2, -1, -1, 1, -6, -3, -2, -1, -1, 1, -7, -3, -2, -1, -1, -1, 1, -8, -4, -2, -2, -1, -1, -1, 1, -9, -4, -3, -2, -1, -1, -1, -1, 1, -10, -5, -3, -2, -2, -1, -1, -1, -1, 1, -11, -5, -3, -2, -2, -1, -1, -1, -1, -1, 1, -12, -6, -4, -3, -2, -2, -1, -1, -1, -1, -1, 1

Offset: 1

Mats Granvik, Paul D. Hanna, Mar 22 2010

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

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Cf. A010766, A154990.

[k eq n select 1 else -Floor(n/k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 06 2021

Table[If[k==n, 1, -Floor[n/k]], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 06 2021 *)

flatten([[1 if k==n else -(n//k) for k in [1..n]] for n in [1..12]]) # G. C. Greubel, Mar 06 2021

T(n, k) = A010766(n, k)*A154990(n, k).

T(n, k) = -floor(n/k) with T(n, n) = 1. - G. C. Greubel, Mar 06 2021

Showing 1-7 of 7 results.

cp's OEIS Frontend

A163270 First column in matrix inverse of (A047999*A154990).

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A080956 a(n) = (n+1)*(2-n)/2.

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A092582 Triangle read by rows: T(n,k) is the number of permutations p of [n] having length of first run equal to k.

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A145839 Number of 3-compositions of n.

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A155038 Triangle read by rows: T(n,k) is the number of compositions of n with first part k.

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A155031 Triangle T(n, k) = 0 if n==0 (mod k) otherwise -1 with T(n, n) = 1 and T(n, 0) = 0, read by rows.

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