A103406 -id:A103406 - cp's OEIS frontend

A127648 Triangle read by rows: row n consists of n zeros followed by n+1.

1, 0, 2, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15

Offset: 0

Gary W. Adamson, Jan 22 2007

Alternatively, a(n) = k if n+1 is the k-th triangular number and 0 otherwise.

Triangle T(n,k), 0<=k<=n, read by rows, given by (0,0,0,0,0,0,0,0,0,0,...) DELTA (2,-1/2,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 27 2011

			First few rows of the triangle:
  1;
  0, 2;
  0, 0, 3;
  0, 0, 0, 4;
  0, 0, 0, 0, 5;
  0, 0, 0, 0, 0, 6;
  0, 0, 0, 0, 0, 0, 7;
  ...

Antti Karttunen, Rows n = 0..360 of the triangle, flattened (Rows n = 0..100 from G. C. Greubel)

Cf. A003506, A007318, A084938, A010054, A103406, A142150.

a127648 n k = a127648_tabl !! n !! k
a127648_row n = a127648_tabl !! n
a127648_tabl = map reverse $ iterate (\(x:xs) -> x + 1 : 0 : xs) [1]
a127648_list = concat a127648_tabl
-- Reinhard Zumkeller, Jul 13 2013

[k eq n select n+1 else 0: k in [0..n], n in [0..20]]; // G. C. Greubel, Mar 12 2024

A127648 := proc(n)
    for i from 0 do
        if A000217(i) = n+1 then
            return i ;
        elif A000217(i) >n then
            return 0 ;
        end if;
    end do;
end proc: # R. J. Mathar, Apr 23 2013

Flatten[Table[{n,Table[0,{n}]},{n,15}]] (* Harvey P. Dale, Jul 27 2011 *)

A127648(n) = if(ispolygonal(1+n,3), (sqrtint(1+((1+n)*8))-1)/2, 0); \\ Antti Karttunen, Jan 19 2025

for i in range(1,15):
    print(i, end=", ")
    for j in range(i):
        print("0", end=", ") # Mohammad Saleh Dinparvar, May 11 2020

from math import isqrt
from sympy.ntheory.primetest import is_square
def A127648(n): return (m:=isqrt(k:=n<<1))+(k>m*(m+1)) if is_square((n<<3)+1) else 0 # Chai Wah Wu, Jun 09 2025

def A127648(n): return (sqrt(9+8*n)-1)//2 if ((sqrt(9+8*n)-3)/2).is_integer() else 0
[A127648(n) for n in range(153)] # G. C. Greubel, Mar 12 2024

Infinite lower triangular matrix with (1, 2, 3, ...) in the main diagonal and the rest zeros.
This sequence * A007318 (Pascal's Triangle) = A003506.
A007318 * this sequence = A103406.
G.f.: 1/(x*y-1)^2. - R. J. Mathar, Aug 11 2015
a(n) = (1/2) (round(sqrt(4 + 2 n)) - round(sqrt(2 + 2 n))) (-1 + round(sqrt(2 + 2 n)) + round(sqrt(4 + 2 n))). - Brian Tenneson, Jan 27 2017
From G. C. Greubel, Mar 13 2024: (Start)
T(n, n) = n+1.
Sum_{k=0..n} T(n, k) = n+1.
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A142150(n+2).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^floor(n/2)*A142150(n+2). (End)

A128064 Triangle T with T(n,n)=n, T(n,n-1)=-(n-1) and otherwise T(n,k)=0; 0

Original entry on oeis.org

1, -1, 2, 0, -2, 3, 0, 0, -3, 4, 0, 0, 0, -4, 5, 0, 0, 0, 0, -5, 6, 0, 0, 0, 0, 0, -6, 7, 0, 0, 0, 0, 0, 0, -7, 8, 0, 0, 0, 0, 0, 0, 0, -8, 9, 0, 0, 0, 0, 0, 0, 0, 0, -9, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 12

Offset: 1

Gary W. Adamson, Feb 14 2007

The matrix inverse = (1/1; 1/2, 1/2; 1/3, 1/3, 1/3;...). Binomial transform of A128064 = A128065. A128064 * A007318 = A103406.
The positive version with row sums 2n+1 is given by T(n,k)=sum{j=k..n, C(n,j)*C(j,k)*(-1)^(n-j)*(j+1)}. - Paul Barry, May 26 2007
Binomial transform of unsigned sequence is A003506. - Gary W. Adamson, Aug 29 2007
Table T(n,k) read by antidiagonals. T(n,1) = n (for n>1), T(n,2) = -n, T(n,k) = 0, k > 2. - Boris Putievskiy, Feb 07 2013

			First few rows of the triangle are:
1;
-1,2;
0,-2,3;
0,0,-3,4;
0,0,0,-4,5;
0,0,0,0,-5,6;
0,0,0,0,0,-6,7;
...
From _Boris Putievskiy_, Feb 07 2013: (Start)
The start of the sequence as table:
1..-1..0..0..0..0..0...
2..-2..0..0..0..0..0...
3..-3..0..0..0..0..0...
4..-4..0..0..0..0..0...
5..-5..0..0..0..0..0...
6..-6..0..0..0..0..0...
7..-7..0..0..0..0..0...
. . .
(End)

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A128065, A103406, A003506, A002260, A167374.

row[1] = {1}; row[2] = {-1, 2}; row[n_] := Join[Array[0&, n-2], {-n+1, n}]; Table[row[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jan 12 2015 *)

Number triangle T(n,k)=sum{j=k..n, C(n,j)*C(j,k)*(-1)^(j-k)*(j+1)}. - Paul Barry, May 26 2007
a(n) = A002260(n)*A167374(n); a(n) = i*floor((i+2)/(t+2))*(-1)^(i+t+1), where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Feb 07 2013
G.f.: (-1)^k*[x^k*exp(k*x)]'/exp(k*x)=sum(n>=k, (-1)^n*T(n,k)*x^n). - Vladimir Kruchinin, Oct 18 2013

A154286 a(n) = E(k)C(n+k,k) = Euler(k)binomial(n+k,k) for k=4.

Original entry on oeis.org

5, 25, 75, 175, 350, 630, 1050, 1650, 2475, 3575, 5005, 6825, 9100, 11900, 15300, 19380, 24225, 29925, 36575, 44275, 53130, 63250, 74750, 87750, 102375, 118755, 137025, 157325, 179800, 204600, 231880, 261800, 294525, 330225, 369075, 411255

Offset: 0

Peter Luschny, Jan 06 2009

a(n) = E(4)*binomial(n+4,4) where E(n) are the Euler number in the enumeration A122045.
a(n) is the special case k=4 in the sequence of diagonals in the triangle A153641.
a(n) is the 5th row in A093375.
a(n) is the 5th column in A103406.
a(n) is the 5th antidiagonal in A103283.
(a(n+1) - a(n))/5 are the pyramidal numbers A000292 (n>1).
(a(n+2) - 2a(n+1) + a(n))/5 are the triangular numbers A000217 (n>2).
(a(n+3) - 3a(n+2) + 3a(n+1) - a(n))/5 are the natural numbers A000027 (n > 3).
Number of orbits of Aut(Z^7) as function of the infinity norm (n+4) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 107520. - Philippe A.J.G. Chevalier, Dec 28 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A153641, A000579.

[(n+1)*(n+2)*(n+3)*(n+4)*5 div 24: n in [0..40]]; // Vincenzo Librandi, Sep 10 2016

Maple
```
seq(euler(4)*binomial(n+4,4),n=0..32);
```

CoefficientList[Series[-5/(x - 1)^5, {x, 0, 35}], x] (* Robert G. Wilson v, Jan 29 2015 *)
Table[(n + 1)*(n + 2)*(n + 3)*(n + 4)*5/24, {n, 0, 25}] (* G. C. Greubel, Sep 09 2016 *)
LinearRecurrence[{5,-10,10,-5,1},{5,25,75,175,350},40] (* Harvey P. Dale, Nov 18 2021 *)

x='x+O('x^99); Vec(5/(1-x)^5) \\ Altug Alkan, Sep 10 2016

a(n) = (n+1)*(n+2)*(n+3)*(n+4)*5/24.
a(n) = a(n-1)*(n+4)/n (n>0), a(0)=5.
O.g.f.: 5/(1-x)^5.
E.g.f.: (5/24)*x*(24 + 36*x + 12*x^2 + x^3)*exp(x). - G. C. Greubel, Sep 09 2016
a(n) = 5*A000332(n+4). - Michel Marcus, Sep 10 2016

A093375 Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 6, 1, 5, 16, 18, 8, 1, 6, 25, 40, 30, 10, 1, 7, 36, 75, 80, 45, 12, 1, 8, 49, 126, 175, 140, 63, 14, 1, 9, 64, 196, 336, 350, 224, 84, 16, 1, 10, 81, 288, 588, 756, 630, 336, 108, 18, 1, 11, 100, 405, 960, 1470, 1512, 1050, 480, 135, 20, 1, 12

Offset: 1

Ralf Stephan, Apr 28 2004

Number of n-long m-ary words avoiding the pattern 1-1'2'.
T(n,n+1) = Sum_{i=1..n} T(n,i).
Exponential Riordan array [(1+x)e^x, x] as a number triangle. - Paul Barry, Feb 17 2009
From Peter Bala, Jul 22 2014: (Start)
Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/
having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A059298. (End)

			Array T(m,n) (with rows m >= 1 and columns n >= 1) begins as follows:
   1   1   1   1   1   1 ...
   2   4   6   8  10  12 ...
   3   9  18  30  45  63 ...
   4  16  40  80 140 224 ...
   5  25  75 175 350 630 ...
   ...
Triangle S(n,k) = T(n-k+1, k+1) begins
.n\k.|....0....1....2....3....4....5....6
= = = = = = = = = = = = = = = = = = = = =
..0..|....1
..1..|....2....1
..2..|....3....4....1
..3..|....4....9....6....1
..4..|....5...16...18....8....1
..5..|....6...25...40...30...10....1
..6..|....7...36...75...80...45...12....1
...

Muniru A Asiru, Antidiagonals, n=1..100 flattened
Yasemin Alp and E. Gokcen Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 7.
Sergey Kitaev and Toufik Mansour, Partially ordered generalized patterns and k-ary words, arXiv:math/0210023 [math.CO], 2002.
Wikipedia, Sheffer sequence.

Rows include A045943. Columns include A002411, A027810.
Main diagonal is A037965. Subdiagonals include A002457.
Antidiagonal sums are A001792.
See A103283 for a signed version.
Cf. A103406, A059298, A073107 (unsigned inverse).

nmax:=14;; T:=List([1..nmax],n->List([1..nmax],k->k*Binomial(n+k-2,n-1)));;
b:=List([2..nmax],n->OrderedPartitions(n,2));;
a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->T[b[i][j][1]][b[i][j][2]]))); # Muniru A Asiru, Aug 07 2018

nmax = 10;
T = Transpose[CoefficientList[# + O[z]^(nmax+1), z]& /@ CoefficientList[(1 - x z)/(1 - z - x z)^2 + O[x]^(nmax+1), x]];
row[n_] := T[[n+1, 1 ;; n+1]];
Table[row[n], {n, 0, nmax}] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

# uses[riordan_array from A256893]
riordan_array((1+x)*exp(x), x, 8, exp=true) # Peter Luschny, Nov 02 2019

Triangle = P*M, the binomial transform of the infinite bidiagonal matrix M with (1,1,1,...) in the main diagonal and (1,2,3,...) in the subdiagonal, and zeros elsewhere. P = Pascal's triangle as an infinite lower triangular matrix. - Gary W. Adamson, Nov 05 2006
From Peter Bala, Sep 20 2012: (Start)
E.g.f. for triangle: (1 + z)*exp((1 + x)*z) = 1 + (2 + x)*z + (3 + 4*x + x^2)*z^2/2! + ....
O.g.f. for triangle: (1 - x*z)/(1 - z - x*z)^2 = 1 + (2 + x)*z + (3 + 4*x + x^2)*z^2 + ....
The n-th row polynomial R(n,x) of the triangle equals (1+x)^n + n*(1+x)^(n-1) for n >= 0 and satisfies d/dx(R(n,x)) = n*R(n-1,x), as well as R(n,x+y) = Sum_{k = 0..n} binomial(n,k)*R(k,x)*y^(n-k). The row polynomials are a Sheffer sequence of Appell type.
Matrix inverse of the triangle is a signed version of A073107. (End)
From Tom Copeland, Oct 20 2015: (Start)
With offset 0 and D = d/dx, the raising operator for the signed row polynomials P(n,x) is RP = x - d{log[e^D/(1-D)]}/dD = x - 1 - 1/(1-D) =  x - 2 - D - D^2 + ..., i.e., RP P(n,x) = P(n+1,x).
The e.g.f. for the signed array is (1-t) * e^(-t) * e^(x*t).
From the Appell formalism, the row polynomials PI(n,x) of A073107 are the umbral inverse of this entry's row polynomials; that is, P(n,PI(.,x)) = x^n = PI(n,P(.,x)) under umbral composition. (End)
From Petros Hadjicostas, Nov 01 2019: (Start)
As a triangle, we let S(n,k) = T(n-k+1, k+1) = (n-k+1)*binomial(n, k) for n >= 0 and 0 <= k <= n. See the example below.
As stated above by Peter Bala, Sum_{n,k >= 0} S(n,k)*z^n*x^k = (1 - x*z)/(1 - z -x*z)^2.
Also, Sum_{n, k >= 0} S(n,k)*z^n*x^k/n! = (1+z)*exp((1+x)*z).
As he also states, the n-th row polynomial is R(n,x) = Sum_{k = 0..n} S(n, k)*x^k = (1 + x)^n + n*(1 + x)^(n-1).
If we define the signed triangle S*(n,k) = (-1)^(n+k) * S(n,k) = (-1)^(n+k) * T(n-k+1, k+1), as Tom Copeland states, Sum_{n,k >= 0} S^*(n,k)*t^n*x^k/n! = (1-t)*exp((1-x)*(-t)) = (1-t) * e^(-t) * e^(x*t).
Apparently, S*(n,k) = A103283(n,k).
As he says above, the signed n-th row polynomial is P(n,x) =  (-1)^n*R(n,-x) = (x - 1)^n - n*(x - 1)^(n-1).
According to Gary W. Adamson, P(n,x) is "the monic characteristic polynomial of the n X n matrix with 2's on the diagonal and 1's elsewhere." (End)

A103283 Triangle read by rows: T(n,k) is the coefficient of x^k in the monic characteristic polynomial of the n X n matrix with 2's on the diagonal and 1's elsewhere (n >= 1 and 0 <= k <= n). Row 0 consists of the single term 1.

Original entry on oeis.org

1, -2, 1, 3, -4, 1, -4, 9, -6, 1, 5, -16, 18, -8, 1, -6, 25, -40, 30, -10, 1, 7, -36, 75, -80, 45, -12, 1, -8, 49, -126, 175, -140, 63, -14, 1, 9, -64, 196, -336, 350, -224, 84, -16, 1, -10, 81, -288, 588, -756, 630, -336, 108, -18, 1, 11, -100, 405, -960, 1470, -1512, 1050, -480, 135, -20, 1, -12, 121, -550, 1485, -2640, 3234, -2772, 1650, -660, 165, -22, 1

Offset: 0

Gary W. Adamson, Feb 04 2005

			The monic characteristic polynomial of the matrix [2 1 1 / 1 2 1 / 1 1 2] is x^3 - 6*x^2 + 9*x - 4; so T(3,0) = -4, T(3,1) = 9, T(3,2) = -6, T(3,3) = 1.
Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
   1;
  -2,   1;
   3,  -4,  1;
  -4,   9, -6,  1;
   5, -16, 18, -8, 1;
   ...

Row sums yield the sequence 1, -1, 0, 0, 0, ... . Row sums of the unsigned triangle yield A001792. See A093375 for the unsigned version. A103406 is a mirror image.

with(linalg): a:=proc(i,j) if i=j then 2 else 1 fi end: 1;for n from 1 to 11 do seq(coeff(expand(x*charpoly(matrix(n,n,a),x)),x^k),k=1..n+1) od; # yields the sequence in triangular form

M[n_] := IdentityMatrix[n] + 1;
row[n_] := row[n] = If[n == 0, {1}, If[OddQ[n], -1, 1]* CharacteristicPolynomial[M[n], x] // CoefficientList[#, x]&];
T[n_, k_] := row[n][[k + 1]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 30 2024 *)

O.g.f.: (1 - x*y)/(1 - x*y + y)^2 = 1 + (-2 + x)*y + (3 - 4*x + y^2)*y^2 + .... - Peter Bala, Oct 18 2023

Edited by Emeric Deutsch, Mar 19 2005

A103407 Triangle of absolute values of the coefficients (in descending powers) of the characteristic polynomials of n X n matrices with 3's on the main diagonal and 1's elsewhere.

Original entry on oeis.org

1, 1, 3, 1, 6, 8, 1, 9, 24, 20, 1, 12, 48, 80, 48, 1, 15, 80, 200, 240, 112, 1, 18, 120, 400, 720, 672, 256, 1, 21, 168, 700, 1680, 2352, 1792, 576, 1, 24, 224, 1120, 3360, 6272, 7168, 4608, 1280, 1, 27, 288, 1680, 6048, 14112, 21504, 20736, 11520, 2816, 1, 30

Offset: 0

Gary W. Adamson, Feb 04 2005

			3rd row (1, 9, 24, 20) with alternating signs = characteristic polynomial 3 X 3 matrix [3 1 1 / 1 3 1 / 1 1 3], x^3 - 9x^2 + 24x - 20.

Row sums are A006234: 1, 4, 15, 54, 189... Rightmost terms in each row = A001792: 1, 3, 8, 20, 48, 112, 256...(row sums of A103406, the analogous triangle with all 2's in the generating matrix.)

See A103247 for another version.

Extended and edited by John W. Layman, Mar 17 2005

A128065 Binomial transform of A128064.

Original entry on oeis.org

1, 0, 2, -1, 2, 3, -2, 0, 6, 4, -3, -4, 6, 12, 5, -4, -10, 0, 20, 20, 6, -5, -18, -15, 20, 45, 30, 7, -6, -28, -42, 0, 70, 84, 42, 8, -7, -40, -84, -56, 70, 168, 140, 56, 9, -8, -54, -144, -168, 0, 252, 336, 216, 72, 10

Offset: 1

Gary W. Adamson, Feb 14 2007

A128064 * A007318 = A103406 row sums = 2^(n-1).

			First few rows of the triangle are:
   1;
   0,  2;
  -1,  2,  3;
  -2,  0,  6,  4;
  -3, -4,  6, 12,  5;
  ...

Cf. A128064, A103406.

/* As triangle */ [[(k+1)*(Binomial(n,k)-Binomial(n,k+1)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 12 2019

Table[(k + 1) (Binomial[n, k] - Binomial[n, k + 1]), {n, 0,
12}, {k, 0, n}] // Flatten (* Nathaniel MacFadden, Jul 11 2019 *)

A007318 * A128064 as infinite lower triangular matrices.

a(43) corrected by Nathaniel MacFadden, Jul 11 2019

A347056 Triangle read by rows: T(n,k) = (n+1)(n+2)(k+3)*binomial(n,k)/6, 0 <= k <= n.

Original entry on oeis.org

1, 3, 4, 6, 16, 10, 10, 40, 50, 20, 15, 80, 150, 120, 35, 21, 140, 350, 420, 245, 56, 28, 224, 700, 1120, 980, 448, 84, 36, 336, 1260, 2520, 2940, 2016, 756, 120, 45, 480, 2100, 5040, 7350, 6720, 3780, 1200, 165, 55, 660, 3300, 9240, 16170, 18480, 13860, 6600, 1815, 220

Offset: 0

Luc Rousseau, Aug 14 2021

This triangle is T[3] in the sequence (T[p]) of triangles defined by: T[p](n,k) = (k+p)*(n+p-1)!/(k!*(n-k)!*p!) and T[0](0,0)=1.

Riordan triangle (1/(1-x)^3, x/(1-x)) with column k scaled with A000292(k+1) = binomial(k+3, 3), for k >= 0. - Wolfdieter Lang, Sep 30 2021

			T(6,2) = (6+1)*(6+2)*(2+3)*binomial(6,2)/6 = 7*8*5*15/6 = 700.
The triangle T begins:
n \ k  0   1    2     3     4     5     6     7     8    9  10 ...
0:     1
1:     3   4
2:     6  16   10
3:    10  40   50    20
4:    15  80  150   120    35
5:    21 140  350   420   245    56
6:    28 224  700  1120   980   448    84
7:    36 336 1260  2520  2940  2016   756   120
8:    45 480 2100  5040  7350  6720  3780  1200   165
9:    55 660 3300  9240 16170 18480 13860  6600  1815  220
10:   66 880 4950 15840 32340 44352 41580 26400 10890 2640 286
... - _Wolfdieter Lang_, Sep 30 2021

Luc Rousseau, Illustration: the A347056 triangle in a sequence of contiguous triangles.

Cf. A097805 (p=0), A103406 (p=1), A124932 (essentially p=2).
From Wolfdieter Lang, Sep 30 2021: (Start)
Columns (with leading zeros): A000217(n+1), 4*A000294, 10*A000332(n+2), 20*A000389(n+2), 35*A000579(n+2), 56*A000580(n+2), 84*A000581(n+2), 120*A000582(n+2), ...
Diagonals: A000292(k+1), A004320(k+1), 2*A006411(k+1), 10*A040977, ... (End)

T(p,n,k)=if(n==0&&p==0,1,((k+p)*(n+p-1)!)/(k!*(n-k)!*p!))
for(n=0,9,for(k=0,n,print1(T(3,n,k),", ")))

T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6.

G.f. column k: x^k*binomial(k+3, 3)/(1 - x)^(k+3), for k >= 0. - Wolfdieter Lang, Sep 30 2021

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A127648 Triangle read by rows: row n consists of n zeros followed by n+1.

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A128064 Triangle T with T(n,n)=n, T(n,n-1)=-(n-1) and otherwise T(n,k)=0; 0

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A154286 a(n) = E(k)*C(n+k,k) = Euler(k)*binomial(n+k,k) for k=4.

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Magma

Maple

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PARI

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A093375 Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.

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GAP

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Sage

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A103283 Triangle read by rows: T(n,k) is the coefficient of x^k in the monic characteristic polynomial of the n X n matrix with 2's on the diagonal and 1's elsewhere (n >= 1 and 0 <= k <= n). Row 0 consists of the single term 1.

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Maple

Mathematica

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A103407 Triangle of absolute values of the coefficients (in descending powers) of the characteristic polynomials of n X n matrices with 3's on the main diagonal and 1's elsewhere.

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A128065 Binomial transform of A128064.

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A154286 a(n) = E(k)C(n+k,k) = Euler(k)binomial(n+k,k) for k=4.

A347056 Triangle read by rows: T(n,k) = (n+1)(n+2)(k+3)*binomial(n,k)/6, 0 <= k <= n.