A099567 -id:A099567 - cp's OEIS frontend

A144904 Coefficient of x^n in expansion of x/((1-x-x^3)*(1-x)^(n-1)), also diagonal of A144903.

0, 1, 2, 6, 21, 76, 280, 1045, 3937, 14938, 56993, 218414, 840090, 3241153, 12537263, 48604755, 188799962, 734631798, 2862843281, 11171582151, 43647688211, 170720728344, 668414462009, 2619400928928, 10273572796046, 40325085206853, 158393604268277

Offset: 0

Alois P. Heinz, Sep 24 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A099567, A144903.

A144904:= func< n | n eq 0 select 0 else  (&+[Binomial(2*n-2*j-2, n+j-1): j in [0..Floor((n-1)/3)]]) >;
[A144904(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

A:= proc(n,k) coeftayl (x/ (1-x-x^3)/ (1-x)^(k-1), x=0, n) end:
a:= n-> A(n,n):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, n,
      ((27*n^3-150*n^2+195*n-12)*a(n-1)
      -(66*n^3-382*n^2+492*n+124)*a(n-2)
      +(27*n^3-156*n^2+201*n+48)*a(n-3)
      -2*(2*n-7)*(3*n^2-7*n-2)*a(n-4))/((n-1)*(3*n^2-13*n+8)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 06 2013

Table[Sum[Binomial[2*n-2*j-2, n+j-1], {j,0,Floor[(n-1)/3]}], {n,0,40}] (* G. C. Greubel, Jul 27 2022 *)

def A144904(n): return sum(binomial(2*n-2*j-2, n+j-1) for j in (0..((n-1)//3)))
[A144904(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

a(n) = [x^n] x/((1-x-x^3)*(1-x)^(n-1)).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n-1)/3)} binomial(2*n-2*j-2, n+j-1).
a(n) = A099567(2*n, n). (End)
a(n) = binomial(2*(n-1), n-1)*hypergeom([1, (1-n)/3, (2-n)/3, 1-n/3], [1-n, 3/2-n, n], -27/4) for n > 0. - Stefano Spezia, Apr 06 2024
a(n) ~ 4^n/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 08 2024

A077868 Expansion of 1/((1-x)*(1-x-x^3)).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 18, 27, 40, 59, 87, 128, 188, 276, 405, 594, 871, 1277, 1872, 2744, 4022, 5895, 8640, 12663, 18559, 27200, 39864, 58424, 85625, 125490, 183915, 269541, 395032, 578948, 848490, 1243523, 1822472, 2670963, 3914487, 5736960, 8407924, 12322412

Offset: 0

N. J. A. Sloane, Nov 17 2002

Row sums of Riordan array (1/(1-x), x*(1+x^2)). - Paul Barry, Feb 16 2005

a(n) is the number of partitions of {1, ..., n+3} into two blocks in which only 1- or 3-strings of consecutive integers can appear in a block and there is at least one 3-string. E.g., a(3)=5 because the enumerated partitions of {1,2,3,4,5,6} are 1235/46, 1345/26, 15/2346, 13/2456, 123/456. - Augustine O. Munagi, Apr 11 2005

Chu, Hung Viet. "Various Sequences from Counting Subsets." Fib. Quart., 59:2 (May 2021), 150-157.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kassie Archer and Aaron Geary, Powers of permutations that avoid chains of patterns, arXiv:2312.14351 [math.CO], 2023. See p. 15.
Hung Viet Chu, Various sequences from counting subsets, arXiv:2005.10081 [math.CO], 2020-2021.
A. O. Munagi, Set Partitions with Successions and Separations, IJMMS 2005:3 (2005), 451-463.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1).

Cf. A000071, A077888, A077941, A105489.
Cf. A000930, A050228, A144898, A144899, A144900, A144901, A144902, A144903, A144904, A226405.
Cf. A078012, A099567, A135851.

A077868:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+, j+1): j in [0..Floor((n+1)/3)]]) >;
[A077868(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

a:= n-> (Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [2,-1,1,-1][i] else 0 fi)^n)[1,1]: seq(a(n), n=0..41); # Alois P. Heinz, Sep 05 2008
g:=(1+z+z^2)/(1-z-z^3): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-1, n=1..42); # Zerinvary Lajos, Jan 09 2009

LinearRecurrence[{1,1,0,0,-1}, {1,2,3,5,8,12}, 42] (* or *)
CoefficientList[Series[1/((1-x)(1-x-x^3)), {x, 0, 41}], x] (* Michael De Vlieger, Jun 06 2018 *)

Vec(1/(1-x)/(1-x-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

{a = vector(50);
a[1] = 1; a[2] = 2; a[3] = 3;
for(n=4,50,
a[n] = 1 + a[n-1] + a[n-3];
); a} \\ Gerry Martens, Jun 03 2018

{a(n) = if( n<0, n=-4-n; polcoeff( -1 / (1 - x) / (1 + x^2 - x^3) + x * O(x^n), n), polcoeff( 1 / (1 - x) / (1 - x - x^3) + x * O(x^n), n))}; /* Michael Somos, Jun 17 2018 */

def A077868(n): return sum(binomial(n-2*j+1, j+1) for j in (0..((n+1)//3)))
[A077868(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

Partial sums of A000930. a(n-1) = Sum_{k=0..floor(n/2)} binomial(n-2*k, k+1). - Paul Barry, Jul 07 2004
a(n-3) = Sum(binomial(n-r, r)), r=1, 2, ... which is the case t=3 and k=2 in the general case of t-strings and k blocks: a(n-3, k, t) = Sum(binomial(n-r*(t-1), r)*S2(n-r*(t-1)-1, k-1)), r=1, 2, ... - Augustine O. Munagi, Apr 11 2005
From Paul Weisenhorn, Oct 28 2011: (Start)
a(n) = a(n-1) + a(n-2) - a(n-5) for n > 4.
a(n) = a(n-2) + a(n-3) + a(n-4) + 2 for n > 3.
G.f.: 1/((1-x)*(1-x-x^3)). (End)
a(n) = 1 + a(n-1) + a(n-3), a(1)=1, a(2)=2, a(3)=3. - Gerry Martens, Jun 10 2018
a(n) = -A077888(-4-n) for all n in Z. - Michael Somos, Jun 17 2018
a(n) = A000930(n+3) - 1. - Greg Dresden, Jun 20 2021
a(n) = A099567(n+3, 4). - G. C. Greubel, Jul 27 2022

More terms from Augustine O. Munagi, Apr 11 2005

A050228 a(n) is the number of subsequences {s(k)} of {1,2,3,...n} such that s(k+1)-s(k) is 1 or 3.

Original entry on oeis.org

1, 3, 6, 11, 19, 31, 49, 76, 116, 175, 262, 390, 578, 854, 1259, 1853, 2724, 4001, 5873, 8617, 12639, 18534, 27174, 39837, 58396, 85596, 125460, 183884, 269509, 394999, 578914, 848455, 1243487, 1822435, 2670925, 3914448, 5736920, 8407883

Offset: 1

John W. Layman, Dec 20 1999

The second differences c(n) of {a(n)} satisfy c(n)=c(n-1)+c(n-3) and give A000930 with the first 5 terms deleted.

Partial sums of A077868. - Paul Barry, Sep 16 2004

Chu, Hung Viet. "Various Sequences from Counting Subsets." Fib. Quart., 59:2 (May 2021), 150-157.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Hung Viet Chu, Various sequences from counting subsets, arXiv:2005.10081 [math.CO], 2020-2021.
Z. Kasa, On scattered subword complexity, arXiv preprint arXiv:1104.4425 [cs.DM], 2011.
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-2,1).

Cf. A000930, A077868, A144898, A144899, A144900, A144901, A144902, A144903, A144904, A226405.

Cf. A078012, A099567, A135851.

A050228:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+1, j+2): j in [0..Floor((n+1)/3)]]) >;
[A050228(n): n in [1..40]]; // G. C. Greubel, Jul 27 2022

with(combstruct): SubSetSeqU := [T, {T=Subst(U,U), S=Set(U, card>=3), U=Sequence(Z, card>=3)}, unlabeled]: seq(count(SubSetSeqU, size=n), n=9..46); # Zerinvary Lajos, Mar 18 2008

Rest[CoefficientList[Series[1/((1-x)^2*(1-x-x^3)), {x, 0, 50}], x]] (* G. C. Greubel, Apr 27 2017 *)
LinearRecurrence[{3,-3,2,-2,1},{1,3,6,11,19},50] (* Harvey P. Dale, Apr 21 2020 *)

my(x='x+O('x^50)); Vec(x/((1-x)^3-x^3*(1-x)^2)) \\ G. C. Greubel, Apr 27 2017

def A050228(n): return sum(binomial(n-2*j+1, j+2) for j in (0..((n+1)//3)))
[A050228(n) for n in (1..40)] # G. C. Greubel, Jul 27 2022

From Paul Barry, Sep 16 2004: (Start)
G.f.: x/((1-x)^3 - x^3(1-x)^2).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 2*a(n-4) + a(n-5).
a(n-1) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k+2). (End)
G.f. = 1/((1-x)^2*(1-x-x^3)). - N. J. A. Sloane, Jun 02 2021
a(n) = A000930(n+5) - n - 4. - Greg Dresden, Jun 20 2021
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+1)/3)} binomial(n-2*j+1, j+2).
a(n) = A099567(n+1, 2). (End)

A144900 Expansion of x/((1-x-x^3)*(1-x)^6).

Original entry on oeis.org

0, 1, 7, 28, 85, 218, 498, 1045, 2055, 3840, 6887, 11945, 20153, 33228, 53741, 85522, 134254, 208344, 320200, 488103, 738951, 1112281, 1666164, 2485845, 3696406, 5481325, 8109676, 11975993, 17658694, 26005706, 38259955, 56243281, 82625979, 121321831, 178067054

Offset: 0

Alois P. Heinz, Sep 24 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,36,-41,36,-27,16,-6,1).

7th column of A144903.

Cf. A099567.

A144900:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+5, j+6): j in [0..Floor((n+5)/3)]]) >;
[A144900(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

a:= n-> (Matrix(9, (i, j)-> if i=j-1 then 1 elif j=1 then [7, -21, 36, -41, 36, -27, 16, -6, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

CoefficientList[Series[x/((1-x-x^3)(1-x)^6), {x,0,40}], x] (* Vincenzo Librandi, Jun 06 2013 *)
LinearRecurrence[{7,-21,36,-41,36,-27,16,-6,1},{0,1,7,28,85,218,498,1045,2055},40] (* Harvey P. Dale, Mar 02 2016 *)

def A144900(n): return sum(binomial(n-2*j+5, j+6) for j in (0..((n+5)//3)))
[A144900(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

G.f.: x/((1-x-x^3)*(1-x)^6).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+5)/3)} binomial(n-2*j+5, j+6).
a(n) = A099567(n+5, 6). (End)

A144901 Expansion of x/((1-x-x^3)*(1-x)^7).

Original entry on oeis.org

0, 1, 8, 36, 121, 339, 837, 1882, 3937, 7777, 14664, 26609, 46762, 79990, 133731, 219253, 353507, 561851, 882051, 1370154, 2109105, 3221386, 4887550, 7373395, 11069801, 16551126, 24660802, 36636795, 54295489, 80301195, 118561150, 174804431, 257430410

Offset: 0

Alois P. Heinz, Sep 24 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,57,-77,77,-63,43,-22,7,-1).

8th column of A144903.

Cf. A099567.

A144901:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+6, j+7): j in [0..Floor((n+6)/3)]]) >;
[A144901(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

a:= n-> (Matrix(10, (i, j)-> if i=j-1 then 1 elif j=1 then [8, -28, 57, -77, 77, -63, 43, -22, 7, -1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

CoefficientList[Series[x/((1-x-x^3)(1-x)^7), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)

def A144901(n): return sum(binomial(n-2*j+6, j+7) for j in (0..((n+6)//3)))
[A144901(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

G.f.: x/((1-x-x^3)*(1-x)^7).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+6)/3)} binomial(n-2*j+6, j+7).
a(n) = A099567(n+6, 7). (End)

A144902 Expansion of x/((1-x-x^3)*(1-x)^8).

Original entry on oeis.org

0, 1, 9, 45, 166, 505, 1342, 3224, 7161, 14938, 29602, 56211, 102973, 182963, 316694, 535947, 889454, 1451305, 2333356, 3703510, 5812615, 9034001, 13921551, 21294946, 32364747, 48915873, 73576675, 110213470, 164508959, 244810154, 363371304, 538175735

Offset: 0

Alois P. Heinz, Sep 24 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,85,-134,154,-140,106,-65,29,-8,1).

9th column of A144903.

Cf. A099567.

A144903:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+7, j+8): j in [0..Floor((n+7)/3)]]) >;
[A144903(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

a:= n-> (Matrix(11, (i, j)-> if i=j-1 then 1 elif j=1 then [9, -36, 85, -134, 154, -140, 106, -65, 29, -8, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

CoefficientList[Series[x/((1-x-x^3)(1-x)^8), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)

def A144903(n): return sum(binomial(n-2*j+7, j+8) for j in (0..((n+7)//3)))
[A144903(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

G.f.: x/((1-x-x^3)*(1-x)^8).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+7)/3)} binomial(n-2*j+7, j+8).
a(n) = A099567(n+7, 8). (End)

A144898 Expansion of x/((1-x-x^3)*(1-x)^4).

Original entry on oeis.org

0, 1, 5, 15, 36, 76, 147, 267, 463, 775, 1262, 2011, 3150, 4867, 7438, 11268, 16951, 25358, 37766, 56047, 82945, 122482, 180553, 265798, 390880, 574358, 843432, 1237966, 1816384, 2664311, 3907237, 5729077, 8399372, 12313154, 18049371, 26456513, 38778103

Offset: 0

Alois P. Heinz, Sep 24 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,11,-9,7,-4,1).

5th column of A144903.
Cf. A000930, A050228, A077868, A144899, A144900, A144901, A144902, A144903, A144904, A226405.
Cf. A078012, A099567, A135851.

A144898:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+3, j+4): j in [0..Floor((n+3)/3)]]) >;
[A144898(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

a:= n-> (Matrix(7, (i, j)-> if i=j-1 then 1 elif j=1 then [5, -10, 11, -9, 7, -4, 1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

CoefficientList[Series[ x/((1-x-x^3)(1-x)^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)

def A144898(n): return sum(binomial(n-2*j+3, j+4) for j in (0..((n+3)//3)))
[A144898(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

G.f.: x/((1-x-x^3)*(1-x)^4).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+3)/3)} binomial(n-2*j+3, j+4).
a(n) = A099567(n+3, 4). (End)

A144899 Expansion of x/((1-x-x^3)*(1-x)^5).

Original entry on oeis.org

0, 1, 6, 21, 57, 133, 280, 547, 1010, 1785, 3047, 5058, 8208, 13075, 20513, 31781, 48732, 74090, 111856, 167903, 250848, 373330, 553883, 819681, 1210561, 1784919, 2628351, 3866317, 5682701, 8347012, 12254249, 17983326, 26382698, 38695852, 56745223, 83201736

Offset: 0

Alois P. Heinz, Sep 24 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,21,-20,16,-11,5,-1).

6th column of A144903.
Cf. A000930, A050228, A077868, A144898, A144900, A144901, A144902, A144903, A144904, A226405.
Cf. A078012, A099567, A135851.

A144899:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+4, j+5): j in [0..Floor((n+4)/3)]]) >;
[A144899(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

a:= n-> (Matrix(8, (i, j)-> if i=j-1 then 1 elif j=1 then [6, -15, 21, -20, 16, -11, 5, -1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

CoefficientList[Series[x/((1-x-x^3)(1-x)^5), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)

def A144899(n): return sum(binomial(n-2*j+4, j+5) for j in (0..((n+4)//3)))
[A144899(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

G.f.: x/((1-x-x^3)*(1-x)^5).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+4)/3)} binomial(n-2*j+4, j+5).
a(n) = A099567(n+4, 5). (End)

A226405 Expansion of x/((1-x-x^3)*(1-x)^3).

Original entry on oeis.org

0, 1, 4, 10, 21, 40, 71, 120, 196, 312, 487, 749, 1139, 1717, 2571, 3830, 5683, 8407, 12408, 18281, 26898, 39537, 58071, 85245, 125082, 183478, 269074, 394534, 578418, 847927, 1242926, 1821840, 2670295, 3913782, 5736217, 8407142, 12321590, 18058510, 26466393

Offset: 0

Alois P. Heinz, Jun 06 2013

From Bruno Berselli, Jun 07 2013: (Start)
A050228(n)  = a(n)   -a(n-1), n>0.
A077868(n-1)= a(n) -2*a(n-1)   +a(n-2), n>1.
A000217(n)  = a(n)   -a(n-1)             -a(n-3), n>2.
A000930(n-1)= a(n) -3*a(n-1) +3*a(n-2)   -a(n-3), n>2.
n           = a(n) -2*a(n-1)   +a(n-2)   -a(n-3)   +a(n-4), n>3.
1           = a(n) -3*a(n-1) +3*a(n-2) -2*a(n-3) +2*a(n-4)   -a(n-5), n>4.
0           = a(n) -4*a(n-1) +6*a(n-2) -5*a(n-3) +4*a(n-4) -3*a(n-5) +a(n-6), n>5.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-4,3,-1).

4th column of A144903.
Cf. A000217, A078012, A099567, A135851.
Cf. A000930, A050228, A077868, A144898, A144899, A144900, A144901, A144902, A144903, A144904, A226405.

A226405:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+2, j+3): j in [0..Floor((n+2)/3)]]) >;
[A226405(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

a:= n-> (Matrix(6, (i, j)-> if i=j-1 then 1 elif j=1 then [4, -6, 5, -4, 3, -1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

LinearRecurrence[{4,-6,5,-4,3,-1}, {0,1,4,10,21,40}, 40]  (* Bruno Berselli, Jun 07 2013 *)
CoefficientList[Series[x/((1-x-x^3)*(1-x)^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 28 2017 *)

my(x='x+O('x^50)); Vec(x/((1-x-x^3)*(1-x)^3)) \\ G. C. Greubel, Apr 28 2017

def A226405(n): return sum(binomial(n-2*j+2, j+3) for j in (0..((n+2)//3)))
[A226405(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

G.f.: x/((1-x-x^3)*(1-x)^3).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+2)/3)} binomial(n-2*j+2, j+3).
a(n) = A099567(n+2, 3). (End)

A099569 Riordan array (((1+x)^2 - x^3)/(1+x)^3, 1/(1+x)).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -2, 3, -3, 1, 4, -5, 6, -4, 1, -7, 9, -11, 10, -5, 1, 11, -16, 20, -21, 15, -6, 1, -16, 27, -36, 41, -36, 21, -7, 1, 22, -43, 63, -77, 77, -57, 28, -8, 1, -29, 65, -106, 140, -154, 134, -85, 36, -9, 1, 37, -94, 171, -246, 294, -288, 219, -121, 45, -10, 1, -46, 131, -265, 417, -540, 582, -507, 340, -166, 55, -11, 1

Offset: 0

Paul Barry, Oct 22 2004

Inverse matrix of A099567. Row sums are A099570.

			Rows begin as:
    1;
   -1,   1;
    1,  -2,    1;
   -2,   3,   -3,   1;
    4,  -5,    6,  -4,    1;
   -7,   9,  -11,  10,   -5,   1;
   11, -16,   20, -21,   15,  -6,   1;
  -16,  27,  -36,  41,  -36,  21,  -7,  1;
   22, -43,   63, -77,   77, -57,  28, -8,  1;
  -29,  65, -106, 140, -154, 134, -85, 36, -9,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A076540, A099570 (row sums), A099567.

[n eq 0 select 1 else (-1)^(n+k)*(Binomial(n, k) + Binomial(n-1, k+2)): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 25 2022

C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if;
end proc:
for n from 0 to 10 do
    seq((-1)^(n+k)*(C(n, n-k) + add((i-2)*C(n-i, n-k-i), i = 3..n)), k = 0..n);
end do; # Peter Bala, Mar 21 2018

T[n_, k_]:= (-1)^(n+k)*(Binomial[n, k] + Binomial[n-1, k+2]);
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 25 2022 *)

def A099569(n, k): return 1 if (n==0) else (-1)^(n+k)*(binomial(n, k) +binomial(n-1, k+2))
flatten([[A099569(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 25 2022

Sum_{k=0..n} T(n, k) = A099570(n).
Columns have g.f. ((1+x)^2 - x^3)/(1+x)^3*(x/(1+x))^k.
T(n,k) = (-1)^(n+k)*(binomial(n, n-k) + Sum_{i = 3..n} (i-2)*binomial(n-i,n-k-i)), for 0 <= k <= n, otherwise 0. - Peter Bala, Mar 21 2018
From G. C. Greubel, Jul 25 2022: (Start)
T(n, k) = (-1)^(n+k)*(binomial(n, k) + binomial(n-1, k+2)), with T(0, k) = 1.
T(2*n-1, n-1) = (-1)^n*A076540(n), n >= 1.
T(n, n-1) = -n. (End)

cp's OEIS Frontend

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