A008315 -id:A008315 - cp's OEIS frontend

A008482 Coefficients in expansion of (x-1)*(1+x)^(n-1), n > 0.

0, -1, 1, -1, 0, 1, -1, -1, 1, 1, -1, -2, 0, 2, 1, -1, -3, -2, 2, 3, 1, -1, -4, -5, 0, 5, 4, 1, -1, -5, -9, -5, 5, 9, 5, 1, -1, -6, -14, -14, 0, 14, 14, 6, 1, -1, -7, -20, -28, -14, 14, 28, 20, 7, 1, -1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1, -1, -9, -35, -75, -90, -42, 42

Offset: 0

N. J. A. Sloane

Apart from initial term, same as A112467. - Philippe Deléham, Nov 07 2006
From Christopher Simoni, Feb 11 2019: (Start)
Along the diagonal containing all the natural numbers, where every number is at point T(n,n-1), T(n,k)^2 can be found by adding T(n+1,k), T(n,k-1), T(n-1,k), and T(n,k+1) together. Empirical observation.
Along the diagonal containing all the natural numbers, where every number is at point T(n,n-1), T(n,k)^3 can by found. The following products are added together: T(n-1,k-1)*T(n,k+1)*T(n+1,k); T(n-1,k)*T(n+1,k+1)*(n,k-1); and 3*T(n,k). Empirical observation. (End)

			Triangle begins:
   0
  -1   1
  -1   0   1
  -1  -1   1    1
  -1  -2   0    2    1
  -1  -3  -2    2    3    1
  -1  -4  -5    0    5    4   1
  -1  -5  -9   -5    5    9   5   1
  -1  -6 -14  -14    0   14  14   6   1
  -1  -7 -20  -28  -14   14  28  20   7   1
  -1  -8 -27  -48  -42    0  42  48  27   8   1
  -1  -9 -35  -75  -90  -42  42  90  75  35   9   1
  -1 -10 -44 -110 -165 -132   0 132 165 110  44  10   1
... - _Philippe Deléham_, Nov 29 2013

A. A. Kirillov, Variations on the triangular theme, Amer. Math. Soc. Transl., (2), Vol. 169, 1995, pp. 43-73, see p. 71.

G. C. Greubel, Rows n=0..100 of triangle, flattened
I. Gessel and S. Ree, Lattice paths and Faber polynomials, Advances in Combinatorial Methods and Applications to Probability and Statistics pp 3-13. DOI: 10.1007/978-1-4612-4140-9_1.

Skew analog of Pascal's triangle A007318, central column gives Catalan numbers A000108, essentially same as A037012, except rows are read from left to right (A037012 = - this sequence).

The positive half of this triangle is A008315. - Michael Somos

Table[Binomial[n-1, k-1] -Binomial[n-1, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 11 2019 *)

{T(n, k) = if( n<1, 0, polcoeff( (x-1) * (1+x)^(n-1), k))}; /* Michael Somos, Oct 04 1999 */

A008482(n,k)={if(k>=n-k, if(k>n-k,-A008482(n,n-k),0), k>2, A008482(n-1,k-1)+A008482(n-1,k), k>1, 1-(n-2)*(n-3)\2, k, 2-n, -1)} \\ M. F. Hasler, Feb 11 2019

[0] + flatten([[binomial(n-1, k-1) -binomial(n-1, k) for k in (0..n)] for n in (1..12)]) # G. C. Greubel, Feb 11 2019

T(n, k) = T(n-1, k-1) + T(n-1, k); T(0, 0) = 0, T(1, 0) = -1, T(1, 1) = 1. - Michael Somos, Oct 04 1999
T(n, k) = binomial(n, k-1) - binomial(n, k). - Michael Somos, Oct 04 1999
T(2n+1,n+1) = A000108(n), Catalan numbers. - Philippe Deléham, Nov 29 2013
a(n) = -A037012(n). - Michael Somos, May 24 2015
T(n, k) = -T(n, n - k); T(n, n) = 1 except for n = 0; T(n+1, n) = n-1; T(n+2, n) = A000217(n-1) - 1. This proves the observations by C. Simoni, e.g., T(n+1,n)^2 - 1 = T(n+1,n-1) + T(n+2,n) + 1. - M. F. Hasler, Feb 11 2019

A361887 a(n) = S(5,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

Original entry on oeis.org

1, 1, 2, 33, 276, 4150, 65300, 1083425, 20965000, 399876876, 8461219032, 178642861782, 4010820554664, 90684123972156, 2130950905378152, 50560833176021025, 1231721051614138800, 30294218438009039800, 759645100717216142000, 19213764100954274616908, 493269287121905287769776

Offset: 0

Peter Bala, Mar 28 2023

For r a positive integer define S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r. The present sequence is {S(5,n)}. Gould (1974) conjectured that S(3,n) was always divisible by S(1,n). See A183069 for {S(3,n)/S(1,n)}. In fact, calculation suggests that if r is odd then S(r,n) is always divisible by S(1,n).

a(n) is the total number of 5-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2). - Alois P. Heinz, Apr 02 2023

Seiichi Manyama, Table of n, a(n) for n = 0..673
H. W. Gould, Problem E2384, Amer. Math. Monthly, 81 (1974), 170-171.

Cf. A003161 ( S(3,n) ), A003162 ( S(3,n)/S(1,n) ), A183069 ( S(3,2*n-1)/ S(1,2*n-1) ), A361888 ( S(5,n)/S(1,n) ), A361889 ( S(5,2*n-1)/S(1,2*n-1) ), A361890 ( S(7,n) ), A361891 ( S(7,n)/S(1,n) ), A361892 ( S(7,2*n-1)/S(1,2*n-1) ).
Column k=5 of A357824.
Cf. A008315, A120730.

seq(add( ( binomial(n,k) - binomial(n,k-1) )^5, k = 0..floor(n/2)), n = 0..20);

Table[Sum[(Binomial[n, k] - Binomial[n, k-1])^5, {k,0,Floor[n/2]}], {n,0,20}] (* Vaclav Kotesovec, Aug 27 2023 *)

from math import comb
def A361887(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**5 for j in range((n>>1)+1)) # Chai Wah Wu, Mar 25 2025

a(n) = Sum_{k = 0..floor(n/2)} ( (n - 2*k + 1)/(n - k + 1) * binomial(n,k) )^5.
From Alois P. Heinz, Apr 02 2023: (Start)
a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^5.
a(n) = Sum_{j=0..n} A120730(n,j)^5.
a(n) = A357824(n,5). (End)
a(n) ~ 2^(5*n + 19/2) / (125 * Pi^(5/2) * n^(9/2)). - Vaclav Kotesovec, Aug 27 2023

A361890 a(n) = S(7,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

Original entry on oeis.org

1, 1, 2, 129, 2316, 94510, 4939220, 211106945, 14879165560, 828070125876, 61472962084968, 4223017425122958, 325536754765395096, 25399546083773839692, 2059386837863675003112, 173281152533121109073025, 14789443838781868027714800, 1307994690673355979749969800

Offset: 0

Peter Bala, Mar 30 2023

For r a positive integer define S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r. The present sequence is {S(7,n)}. Gould (1974) conjectured that S(3,n) was always divisible by S(1,n). See A183069 for {S(3,n)/S(1,n)}. In fact, calculation suggests that if r is odd then S(r,n) is always divisible by S(1,n).

a(n) is the total number of 7-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2). - Alois P. Heinz, Apr 02 2023

Seiichi Manyama, Table of n, a(n) for n = 0..483
H. W. Gould, Problem E2384, Amer. Math. Monthly, 81 (1974), 170-171.

Cf. A003161 ( S(3,n) ), A003162 ( S(3,n)/S(1,n) ), A382394 ( S(3,2*n-1) ), A183069 ( S(3,2*n-1)/ S(1,2*n-1) ), A361887 ( S(5,n) ), A361888 ( S(5,n)/S(1,n) ), A361889 ( S(5,2*n-1)/S(1,2*n-1) ), A361891 ( S(7,n)/S(1,n) ), A361892 ( S(7,2*n-1)/ S(1,2*n-1) ).
Column k=7 of A357824.
Cf. A008315, A120730.

seq(add( ( binomial(n,k) - binomial(n,k-1) )^7, k = 0..floor(n/2)), n = 0..20);

Table[Sum[(Binomial[n, k] - Binomial[n, k-1])^7, {k,0,Floor[n/2]}], {n,0,20}] (* Vaclav Kotesovec, Aug 27 2023 *)

from math import comb
def A361890(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**7 for j in range((n>>1)+1)) # Chai Wah Wu, Mar 25 2025

a(n) = Sum_{k = 0..floor(n/2)} ( (n - 2*k + 1)/(n - k + 1) * binomial(n,k) )^7.
From Alois P. Heinz, Apr 02 2023: (Start)
a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^7.
a(n) = Sum_{j=0..n} A120730(n,j)^7.
a(n) = A357824(n,7). (End)
a(n) ~ 3 * 2^(7*n + 27/2) / (2401 * Pi^(7/2) * n^(13/2)). - Vaclav Kotesovec, Aug 27 2023

A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 5, 6, 3, 1, 1, 2, 9, 14, 10, 4, 1, 1, 2, 17, 36, 42, 20, 4, 1, 1, 2, 33, 98, 190, 132, 35, 5, 1, 1, 2, 65, 276, 882, 980, 429, 70, 5, 1, 1, 2, 129, 794, 4150, 7812, 5705, 1430, 126, 6, 1, 1, 2, 257, 2316, 19722, 65300, 78129, 33040, 4862, 252, 6

Offset: 0

Alois P. Heinz, Oct 14 2022

			Square array A(n,k) begins:
  1,  1,   1,    1,     1,       1,        1,         1, ...
  1,  1,   1,    1,     1,       1,        1,         1, ...
  2,  2,   2,    2,     2,       2,        2,         2, ...
  2,  3,   5,    9,    17,      33,       65,       129, ...
  3,  6,  14,   36,    98,     276,      794,      2316, ...
  3, 10,  42,  190,   882,    4150,    19722,     94510, ...
  4, 20, 132,  980,  7812,   65300,   562692,   4939220, ...
  4, 35, 429, 5705, 78129, 1083425, 15105729, 211106945, ...

Alois P. Heinz, Antidiagonals n = 0..121, flattened
Wikipedia, Counting lattice paths

Columns k=0-7 give A008619, A001405, A000108, A003161, A129123, A361887, A382433, A361890.
Rows n=1-5 give: A000012, A007395, A000051, A001550, A074511.
Main diagonal gives A357825.
Cf. A008315, A120730.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
A:= (n, k)-> add(b(n, n-2*j)^k, j=0..n/2):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - 1, y + j], {j, {-1, 1}}]]];
A[n_, k_] := Sum[b[n, n - 2*j]^k, { j, 0, n/2}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Oct 18 2022, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..floor(n/2)} A008315(n,j)^k.

A(n,k) = Sum_{j=0..n} A120730(n,j)^k for k>=1, A(n,0) = A008619(n).

A382433 a(n) = S(6,n), where S(r,n) = Sum_{k=0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

Original entry on oeis.org

1, 1, 2, 65, 794, 19722, 562692, 15105729, 553537490, 18107304842, 716747344436, 27247858130506, 1137502720488532, 47573235297987700, 2085487143991309320, 92820152112054862785, 4246321874111740074210, 197525644801830489637170, 9363425291004877645851300

Offset: 0

Seiichi Manyama, Mar 25 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..562

Column k=6 of A357824.
Cf. A000108, A129123.
Cf. A008315, A120730, A382435.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> add(b(n, n-2*j)^6, j=0..n/2):
seq(a(n), n=0..18);  # Alois P. Heinz, Mar 25 2025

Table[Sum[Binomial[n,k] * (Binomial[n,k] - Binomial[n,k-1])^5, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 25 2025 *)

a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^5);

from math import comb
def A382433(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**6 for j in range((n>>1)+1)) # Chai Wah Wu, Mar 25 2025

a(n) = Sum_{k=0..floor(n/2)} A008315(n,k)^6.
a(n) = Sum_{k=0..n} A120730(n,k)^6.
a(n) = A357824(n,6).
a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^5.
a(n) ~ 5 * 2^(6*n+4) / (3^(5/2) * Pi^(5/2) * n^(11/2)). - Vaclav Kotesovec, Mar 25 2025

A173284 Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 2, 1, 8, 3, 1, 13, 5, 2, 21, 8, 3, 1, 34, 13, 5, 2, 1, 55, 21, 8, 3, 1, 89, 34, 13, 5, 2, 1, 144, 55, 21, 8, 3, 1, 233, 89, 34, 13, 5, 2, 1, 377, 144, 55, 21, 8, 3, 1, 610, 233, 89, 34, 13, 5, 2, 1

Offset: 0

Gary W. Adamson, Feb 14 2010

The row sums equal A052952.
Let the triangle = M. Then lim_{n->infinity} M^n = A173285 as a left-shifted vector.
A173284 * [1, 2, 3, ...] = A054451: (1, 1, 4, 5, 12, 17, 33, ...). - Gary W. Adamson, Mar 03 2010
From Johannes W. Meijer, Sep 05 2013: (Start)
Triangle read by rows formed from antidiagonals of triangle A104762.
The diagonal sums lead to A004695. (End)

			First few rows of the triangle:
    1;
    1;
    2,   1;
    3,   1;
    5,   2,  1;
    8,   3,  1;
   13,   5,  2,  1;
   21,   8,  3,  1;
   34,  13,  5,  2,  1;
   55,  21,  8,  3,  1;
   89,  34, 13,  5,  2, 1;
  144,  55, 21,  8,  3, 1;
  233,  89, 34, 13,  5, 2, 1;
  377, 144, 55, 21,  8, 3, 1;
  610, 233, 89, 34, 13, 5, 2, 1;
  ...

Cf. A000045, A004695, A052952, A054451, A173285.

Cf. (Similar triangles) A008315 (Catalan), A011973 (Pascal), A102541 (Losanitsch), A122196 (Fractal), A122197 (Fractal), A128099 (Pell-Jacobsthal), A152198, A152204, A207538, A209634.

T := proc(n, k): if n<0 then return(0) elif k < 0 or k > floor(n/2) then return(0) else combinat[fibonacci](n-2*k+1) fi: end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..14); # Johannes W. Meijer, Sep 05 2013

Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.
From Johannes W. Meijer, Sep 05 2013: (Start)
T(n,k) = A000045(n-2*k+1), n >= 0 and 0 <= k <= floor(n/2).
T(n,k) = A104762(n-k, k). (End)

Term a(15) corrected by Johannes W. Meijer, Sep 05 2013

A289871 Irregular triangle read by rows T(n, k) is the number of admissible pinnacle sets with maximum element n and cardinality k.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 5, 0, 1, 5, 9, 0, 1, 6, 14, 14, 0, 1, 7, 20, 28, 0, 1, 8, 27, 48, 42, 0, 1, 9, 35, 75, 90, 0, 1, 10, 44, 110, 165, 132, 0, 1, 11, 54, 154, 275, 297, 0, 1, 12, 65, 208, 429, 572, 429, 0, 1, 13, 77, 273, 637, 1001, 1001

Offset: 0

Michel Marcus, Jul 14 2017

See David et al. link for definitions.

			Triangle begins:
1;
0;
0;
0, 1;
0, 1;
0, 1, 2;
0, 1, 3;
0, 1, 4, 5;
0, 1, 5, 9;
...

Robert Davis, Sarah A. Nelson, T. Kyle Petersen, Bridget E. Tenner, The pinnacle set of a permutation, arXiv:1704.05494 [math.CO], 2017.

Cf. A008315, A037952 (row sums).

T[0, 0] = 1; T[, 0] = 0; T[n, k_] /; n <= 2k = 0; T[n_, k_] := T[n, k] = Sum[T[j, k-1], {j, 0, n-1}];
Table[T[n, k], {n, 0, 16}, {k, 0, Max[0, (n-1)/2]}] // Flatten (* Jean-François Alcover, Feb 02 2019 *)

T(n, k) = {if ((n==0) && (k==0), return (1)); if (n <= 2*k, return(0)); sum(kk=0, n-1, T(kk, k-1));}
tabf(nn) = {print(T(0, 0), ", "); for (n=1, nn, for (k=0, round(n-1)\2, print1(T(n, k), ", ");); print(););}

T(n,k) = Sum_{n>2k} T(n,k-1) if n>2*k; T(n,k) = 0 if n<=2*k; T(0,0) = 1.

A357871 Total number of n-multisets of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2).

Original entry on oeis.org

1, 1, 2, 5, 21, 183, 3424, 155833, 25962389, 10152021001, 18355563410823, 94826525443572702, 1720192707342762602561, 135432808172830648285721490, 25492564910167901918236137649748, 28315683468644276652408152922412713937, 65407605920313732627652296139090181364409413

Offset: 0

Alois P. Heinz, Oct 17 2022

nonn

			a(3) = 5:
  {///, ///, ///},
  {//\, //\, //\},
  {//\, //\, /\/},
  {//\, /\/, /\/},
  {/\/, /\/, /\/}.

Alois P. Heinz, Table of n, a(n) for n = 0..63
Vaclav Kotesovec, Graph - the asymptotic ratio (5000 terms)
Wikipedia, Counting lattice paths
Wikipedia, Multiset

Cf. A008315, A357825.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> add(binomial(b(n, n-2*j)+n-1, n), j=0..n/2):
seq(a(n), n=0..16);

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x-1, y+j], {j, {-1, 1}}]]];
a[n_] := Sum[Binomial[b[n, n-2*j]+n-1, n], {j, 0, n/2}];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 17 2022, after Alois P. Heinz *)
Table[Sum[Binomial[Binomial[n,k]*(n-2*k+1)/(n-k+1) + n - 1,n], {k,0,n/2}],{n,0,16}] (* Vaclav Kotesovec, Nov 17 2022 *)

a(n) = Sum_{j=0..floor(n/2)} binomial(A008315(n,j)+n-1,n).
From Vaclav Kotesovec, Nov 19 2022: (Start)
a(n)^(1/n) ~ exp(1/2) * 2^(n + 3/2) / (sqrt(Pi) * n^2).
Limit_{n->infinity} a(n) / (exp(n/2) * 2^(n^2 + 3*n/2) / (Pi^(n/2) * n^(2*n + 1/2))) does not exist, see also graph. (End)

A357825 Total number of n-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j = 0..floor(n/2).

Original entry on oeis.org

1, 1, 2, 9, 98, 4150, 562692, 211106945, 404883552194, 1766902576146876, 40519034229909243476, 2708397617879598970178238, 658332084097982587522119612196, 735037057881394837614680080889845116, 2030001034486747324990010196845670569155080

Offset: 0

Alois P. Heinz, Oct 14 2022

Alois P. Heinz, Table of n, a(n) for n = 0..60
Vaclav Kotesovec, Graph - the asymptotic ratio (5000 terms)
Wikipedia, Counting lattice paths

Main diagonal of A357824.

Cf. A000108, A000225, A008315, A120730, A357871.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> add(b(n, n-2*j)^n, j=0..n/2):
seq(a(n), n=0..15);

Table[Sum[(Binomial[n, k]*(n - 2*k + 1)/(n - k + 1))^n, {k, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 17 2022 *)

a(n) = A357824(n,n).
a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^n.
a(n) = Sum_{j=0..n} A120730(n,j)^n.
a(n) mod 2 = 1 <=> n in { A000225 }.
From Vaclav Kotesovec, Nov 17 2022: (Start)
a(n)^(1/n) ~ exp(-1/2) * 2^(n + 3/2) / (sqrt(Pi)*n).
Limit_{n->oo} a(n) / (2^(n^2 + 3*n/2) / (n^n * exp(n/2) * Pi^(n/2))) does not exist, see also graph. (End)
Conjecture: the superconguence a(2*p-1) == 1 (mod p^3) holds for all primes p >= 5 (checked up to p = 101). - Peter Bala, Mar 20 2023

A152547 Triangle, read by rows, derived from Pascal's triangle (see g.f. and example for generating methods).

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Dec 14 2008

			The number of terms in row n is C(n,[n/2]).
Triangle begins:
[1],
[2],
[3,1],
[4,2,2],
[5,3,3,3,1,1],
[6,4,4,4,4,2,2,2,2,2],
[7,5,5,5,5,5,3,3,3,3,3,3,3,3,3,1,1,1,1,1],
[8,6,6,6,6,6,6,4,4,4,4,4,4,4,4,4,4,4,4,4,4,2,2,2,2,2,2,2,2,2,2,2,2,2,2],
[9,7,7,7,7,7,7,7,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1],
...
ILLUSTRATION OF GENERATING METHOD.
Row n is derived from the binomial coefficients in the following way.
Place markers in an array so that the number of contiguous markers
in row k is C(n,k) and then count the markers along columns.
For example, row 6 of this triangle is generated from C(6,k) like so:
------------------------------------------
1: o - - - - - - - - - - - - - - - - - - -
6: o o o o o o - - - - - - - - - - - - - -
15:o o o o o o o o o o o o o o o - - - - -
20:o o o o o o o o o o o o o o o o o o o o
15:o o o o o o o o o o o o o o o - - - - -
6: o o o o o o - - - - - - - - - - - - - -
1: o - - - - - - - - - - - - - - - - - - -
------------------------------------------
Counting the markers along the columns gives row 6 of this triangle:
[7,5,5,5,5,5,3,3,3,3,3,3,3,3,3,1,1,1,1,1].
Continuing in this way generates all the rows of this triangle.
...
Number of repeated terms in each row of this triangle forms A008315:
1;
1;
1, 1;
1, 2;
1, 3, 2;
1, 4, 5;
1, 5, 9, 5;
1, 6, 14, 14;
1, 7, 20, 28, 14;...

Paul D. Hanna, Table of rows 0..14 listed as n, a(n) for n = 0..7059

Cf. A152548 (row squared sums), A008315; A152545.

{T(n,k)=polcoeff(sum(j=0,n,(x^binomial(n,j) - 1)/(x-1)),k)}
for(n=0,10, for(k=0, binomial(n,n\2)-1, print1(T(n,k),","));print(""))

G.f. of row n: Sum_{k=0..n} (x^binomial(n,k) - 1)/(x-1) = Sum_{k=0..binomial(n,n\2)-1} T(n,k)*x^k.

A152548(n) = Sum_{k=0..C(n,[n/2])-1} T(n,k)^2 = Sum_{k=0..[(n+1)/2]} C(n+1, k)*(n+1-2k)^3/(n+1).

cp's OEIS Frontend

A008482 Coefficients in expansion of (x-1)*(1+x)^(n-1), n > 0.

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A361887 a(n) = S(5,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

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A361890 a(n) = S(7,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

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A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A382433 a(n) = S(6,n), where S(r,n) = Sum_{k=0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

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A173284 Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.

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A289871 Irregular triangle read by rows T(n, k) is the number of admissible pinnacle sets with maximum element n and cardinality k.

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