A103283 -id:A103283 - cp's OEIS frontend

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

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

A103406 Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.

Original entry on oeis.org

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

Offset: 0

Gary W. Adamson, Feb 04 2005

This triangle * [1/1, 1/2, 1/3, ...] = (1, 2, 4, 8, 16, 32, ...). - Gary W. Adamson, Nov 15 2007

Triangle read by rows: T(n,k) = (k+1)*binomial(n,k), 0 <= k <= n. - Philippe Deléham, Apr 20 2009

			Characteristic polynomial of 3 X 3 matrix [2 1 1 / 1 2 1 / 1 1 2] = x^3 - 6x^2 + 9x - 4.
The first few characteristic polynomials are:
  1
  x - 2
  x^2 - 4x + 3
  x^3 - 6x^2 + 9x - 4
  x^4 - 8x^3 + 18x^2 - 16x + 5

Vincenzo Librandi, Rows n = 0..100, flattened

Row sums = A001792: 1, 3, 8, 20, 48, 112, ...
See A103283 for the mirror image.
Cf. A093375, A127648, A128064.

with(linalg): printf(`%d,`,1): for n from 1 to 15 do mymat:=array(1..n, 1..n): for i from 1 to n do for j from 1 to n do if i=j then mymat[i,j]:=2 else mymat[i,j]:=1 fi: od: od: temp:=charpoly(mymat,x): for j from n to 0 by -1 do printf(`%d,`,abs(coeff(temp, x, j))) od: od: # James Sellers, Apr 22 2005
p := (n,x) -> (x+1)^(n-1)+(x+1)^(n-2)*(n-1);
seq(seq(coeff(p(n,x),x,n-j-1),j=0..n-1),n=1..11); # Peter Luschny, Feb 25 2014

t[n_, k_] := (k+1)*Binomial[n, k]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 09 2012, after Philippe Deléham *)

Binomial transform of A127648. - Gary W. Adamson, Nov 15 2007
Equals A128064 * A007318. - Gary W. Adamson, Jan 03 2008
T(n,k) = (k+1)*A007318(n,k). - Philippe Deléham, Apr 20 2009
T(n,k) = Sum_{i=1..k+1} i*binomial(k+1,i)*binomial(n-k,k+1-i). - Mircea Merca, Apr 11 2012
O.g.f.: (1 - y)/(1 - y - x*y)^2 = 1 + (1 + 2*x)*y + (1 + 4*x + 3*x*2)*y^2 + .... - Peter Bala, Oct 18 2023

More terms from James Sellers, Apr 22 2005

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)

A103247 Triangle read by rows: T(n,k) is the coefficient of x^k (0<=k<=n) in the monic characteristic polynomial of the n X n matrix with 3's on the diagonal and 1's elsewhere (n>=1). Row 0 consists of the single term 1.

Original entry on oeis.org

1, -3, 1, 8, -6, 1, -20, 24, -9, 1, 48, -80, 48, -12, 1, -112, 240, -200, 80, -15, 1, 256, -672, 720, -400, 120, -18, 1, -576, 1792, -2352, 1680, -700, 168, -21, 1, 1280, -4608, 7168, -6272, 3360, -1120, 224, -24, 1, -2816, 11520, -20736, 21504, -14112, 6048, -1680, 288, -27, 1, 6144, -28160, 57600, -69120, 53760, -28224, 10080, -2400, 360, -30, 1

Offset: 0

Emeric Deutsch, Mar 19 2005

Row sums of the unsigned triangle yield A006234. The unsigned triangle is the mirror image of A103407.

			The monic characteristic polynomial of the matrix [3 1 1 / 1 3 1 / 1 1 3] is x^3 - 9x^2 + 24x - 20; so T(3,0)=-20, T(3,1)=24, T(3,2)=-9, T(3,3)=1.
Triangle begins:
  1;
  -3,1;
  8,-6,1;
  -20,24,-9,1;
  48,-80,48,-12,1;
  ...

Cf. A006234, A103407.

with(linalg): a:=proc(i,j) if i=j then 3 else 1 fi end: 1;for n from 1 to 10 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_] := Table[If[i == j, 3, 1], {i, 1, n}, {j, 1, n}];
P[n_] := P[n] = CharacteristicPolynomial[M[n], x];
row[n_] := row[n] = If[n == 0, {1}, CoefficientList[P[n]/Coefficient[P[n], x, n], x]];
T[n_, k_] := row[n][[k]];
Table[T[n, k], {n, 0, 10}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 06 2024 *)

Appears to be the matrix product (I-S)*P^(-2), where I is the identity, P is Pascal's triangle A007318 and S is A132440, the infinitesimal generator of P. Cf. A055137 (= (I-S)*P) and A103283 (= (I-S)*P^(-1)). - Peter Bala, Nov 28 2011

cp's OEIS Frontend

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

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A103406 Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.

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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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A103247 Triangle read by rows: T(n,k) is the coefficient of x^k (0<=k<=n) in the monic characteristic polynomial of the n X n matrix with 3's on the diagonal and 1's elsewhere (n>=1). Row 0 consists of the single term 1.

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