A060187 -id:A060187 - cp's OEIS frontend

A144431 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n,1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = -1.

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, 2, -2, 1, 1, -3, 2, 2, -3, 1, 1, -4, 7, -8, 7, -4, 1, 1, -5, 9, -5, -5, 9, -5, 1, 1, -6, 16, -26, 30, -26, 16, -6, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, 1, -8, 29, -64, 98, -112, 98, -64, 29, -8, 1, 1, -9, 35, -75, 90, -42, -42, 90, -75, 35, -9, 1

Offset: 1

Roger L. Bagula, Oct 04 2008

Row sums are: {1, 2, 2, 0, 0, 0, 0, 0, 0, 0, ...}.
For m = ...,-1,0,1,2 we get ..., A144431, A007318 (Pascal), A008292, A060187, ..., so this might be called a sub-Pascal triangle.
The triangle starts off like A098593, but is different further on.

			Triangle begins:
  1;
  1,   1;
  1,   0,   1;
  1,  -1,  -1,   1;
  1,  -2,   2,  -2,   1;
  1,  -3,   2,   2,  -3,   1;
  1,  -4,   7,  -8,   7,  -4,   1;
  1,  -5,   9,  -5,  -5,   9,  -5,   1;
  1,  -6,  16, -26,  30, -26,  16,  -6,   1;
  1,  -7,  20, -28,  14,  14, -28,  20,  -7,   1;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 8.

Cf. A007318, A008292, A060187, A098593.

T:=proc(n,k,l) option remember;
if (n=1 or k=1 or k=n) then 1 else
(l*n-l*k+1)*T(n-1,k-1,l)+(l*k-l+1)*T(n-1,k,l); fi; end;
for n from 1 to 15 do lprint([seq(T(n,k,-1),k=1..n)]); od; # N. J. A. Sloane, May 08 2013

m=-1;
T[n_, 1]:= 1; T[n_, n_]:= 1;
T[n_, k_]:= (m*n-m*k+1)*T[n-1, k-1] + (m*k - (m - 1))*T[n-1,k];
Table[T[n, k], {n,15}, {k,n}]//Flatten

def A144431(n,k):
    if (n<3): return 1
    else: return (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3)
flatten([[A144431(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 01 2022

T(n,k) = (m*n - m*k + 1)*T(n-1, k-1) + (m*k - (m-1))*T(n-1, k) with T(n, 1) = T(n, n) = 1 and m = -1.
From G. C. Greubel, Mar 01 2022: (Start)
T(n, n-k) = T(n, k).
T(n, k) = (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3) with T(1, k) = T(2, k) = 1.
Sum_{k=1..n} T(n, k) = [n==1] + 2*[n==2] + 2*[n==3] + (1-(-1)^n)*0^(n-3)*[n>3]. (End)

Edited by N. J. A. Sloane, May 08 2013

A145901 Triangle of f-vectors of the simplicial complexes dual to the permutohedra of type B_n.

Original entry on oeis.org

1, 1, 2, 1, 8, 8, 1, 26, 72, 48, 1, 80, 464, 768, 384, 1, 242, 2640, 8160, 9600, 3840, 1, 728, 14168, 72960, 151680, 138240, 46080, 1, 2186, 73752, 595728, 1948800, 3037440, 2257920, 645120, 1, 6560, 377504, 4612608, 22305024, 52899840

Offset: 0

Peter Bala, Oct 26 2008

The Coxeter group of type B_n may be realized as the group of n X n matrices with exactly one nonzero entry in each row and column, that entry being either +1 or -1. The order of the group is 2^n*n!. The orbit of the point (1,2,...,n) (or any sufficiently generic point (x_1,...,x_n)) under the action of this group is a set of 2^n*n! distinct points whose convex hull is defined to be the permutohedron of type B_n. The rows of this table are the f-vectors of the simplicial complexes dual to these type B permutohedra. Some examples are given in the Example section below. See A060187 for the corresponding table of h-vectors of type B permutohedra.

This is the (unsigned) triangle of connection constants between the polynomial sequences (2*x + 1)^n, n >= 0, and binomial(x+k,k), k >= 0. For example, (2*x + 1)^2 = 8*binomial(x+2,2) - 8*binomial(x+1,1) + 1 and (2*x + 1)^3 = 48*binomial(x+3,3) - 72*binomial(x+2,2) + 26*binomial(x+1,1) - 1. Cf. A163626. - Peter Bala, Jun 06 2019

			The triangle begins
n\k|..0.....1.....2.....3.....4.....5
=====================================
0..|..1
1..|..1.....2
2..|..1.....8.....8
3..|..1....26....72....48
4..|..1....80...464...768...384
5..|..1...242..2640..8160..9600..3840
...
Row 2: the permutohedron of type B_2 is an octagon with 8 vertices and 8 edges. Its dual, also an octagon, has f-vector (1,8,8) - row 3 of this triangle.
Row 3: for an appropriate choice of generic point in R_3, the permutohedron of type B_3 is realized as the great rhombicuboctahedron, also known as the truncated cuboctahedron, with 48 vertices, 72 edges and 26 faces (12 squares, 8 regular hexagons and 6 regular octagons). See the Wikipedia entry and also [Fomin and Reading p.22]. Its dual polyhedron is a simplicial polyhedron, the disdyakis dodecahedron, with 26 vertices, 72 edges and 48 triangular faces and so its f-vector is (1,26,72,48) - row 4 of this triangle.
From _Peter Bala_, Jun 06 2019: (Start)
Examples of falling factorials identities for odd numbered rows: Let (x)_n = x*(x - 1)*...*(x - n + 1) with (x)_0 = 1 denote the falling factorial power.
Row 1: 2*(x)_1 + (0 - 2*x)_1 = 0.
Row 3: 48*(x)_3 + 72*(x)_2 * (2 - 2*x)_1 + 26*(x)_1 * (2 - 2*x)_2 + (2 - 2*x)_3 = 0
Row 5: 3840*(x)_5 + 9600*(x)_4 * (4 - 2*x)_(1) + 8160*(x)_3 * (4 - 2*x)_2 + 2640*(x)_2 * (4 - 2*x)_3 + 242*(x)_1 * (4 - 2*x)_4 + (4 - 2*x)_5 = 0. (End)

Sandrine Dasse-Hartaut and Pawel Hitczenko, Greek letters in random staircase tableaux arXiv:1202.3092v1 [math.CO], 2012.
Peter Bala, Deformations of the Hadamard product of power series
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
S. Fomin and N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004; arXiv:math/0505518 [math.CO], 2005-2008.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
Gábor Hetyei, The type B permutohedron and the poset of intervals as a Tchebyshev transform, University of North Carolina-Charlotte (2019).
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11.
Wikipedia, Truncated cuboctahedron

Cf. A019538 (f-vectors type A permutohedra), A060187 (h-vectors type B permutohedra), A080253 (row sums), A145905, A062715, A028246.
Cf. A258377, A258378, A258379, A258380, A258381.
Cf. A039755, A154537.

with(combinat):
T:= (n,k) -> add((-1)^(k-i)*binomial(k,i)*(2*i+1)^n,i = 0..k):
for n from 0 to 9 do
seq(T(n,k),k = 0..n);
end do;

T[n_, k_] := Sum[(-1)^(k - i)*Binomial[k, i]*(2*i + 1)^n, {i, 0, k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 02 2019 *)

T(n,k) = Sum_{i = 0..k} (-1)^(k-i)*binomial(k,i)*(2*i+1)^n.
Recurrence relation: T(n,k) = (2*k + 1)*T(n-1,k) + 2*k*T(n-1,k-1) with T(0,0) = 1 and T(0,k) = 0 for k >= 1.
Relation with type B Stirling numbers of the second kind: T(n,k) = 2^k*k!*A039755(n,k).
Row sums A080253. The matrix product A060187 * A007318 produces the mirror image of this triangle.
E.g.f.: exp(t)/(1 + x - x* exp(2*t)) = 1 + (1 + 2*x)*t + (1 + 8*x + 8*x^2 )*t^2/2! + ... .
From Peter Bala, Oct 13 2011: (Start)
The polynomials in the first column of the array ((1+t)*P^(-1)-t*P)^(-1), P Pascal's triangle and I the identity, are the row polynomials of this table.
The polynomials in the first column of the array ((1+t)*I-t*A062715)^(-1) are, apart from the initial 1, the row polynomials of this table with an extra factor of t. Cf. A060187. (End)
From Peter Bala, Jul 18 2013: (Start)
Integrating the above e.g.f. with respect to x from x = 0 to x = 1 gives Sum_{k = 0..n} (-1)^k*T(n,k)/(k + 1) = 2^n*Bernoulli(n,1/2), the n-th cosecant number.
The corresponding Type A result is considered in A028246 as Worpitzky's algorithm.
Also for n >= 0, Sum_{k = 0..2*n} (-1)^k*T(2*n,k)/((k + 1)*(k + 2)) = 1/2*2^(2*n)*Bernoulli(2*n,1/2) and for n >= 1, Sum_{k = 0..2*n-1} (-1)^k*T(2*n - 1,k)/((k + 1)*(k + 2)) = -1/2 * 2^(2*n)* Bernoulli(2*n,1/2).
The nonzero cosecant numbers are given by A001896/A001897. (End)
From Peter Bala, Jul 22 2014: (Start)
The row polynomials R(n,x) satisfy the recurrence equation R(n+1,x) = D(R(n,x)) with R(0,x) = 1, where D is the operator 1 + 2*x + 2*x(1 + x)*d/dx.
R(n,x) = 1/(1 + x)* Sum_{k = 0..inf} (2*k + 1)^n*(x/(1 + x))^k, valid for x in the open interval (-1/2, inf). Cf. A019538.
The shifted row polynomial x*R(n,x) = (1 + x)^n*P(n,x/(1 + x)) where P(n,x) denotes the n-th row polynomial of A060187.
The row polynomials R(n,x) have only real zeros.
Symmetry: R(n,x) = (-1)^n*R(n,-1 - x). Consequently the zeros of R(n,x) lie in the open interval (-1, 0). (End)
From Peter Bala, May 28 2015: (Start)
Recurrence for row polynomials: R(n,x) = 1 + x*Sum_{k = 0..n-1} binomial(n,k)2^(n-k)*R(k,x) with R(0,x) = 1.
For a fixed integer k, the expansion of the function A(k,z) := exp( Sum_{n >= 1} R(n,k)*z^n/n ) has integer coefficients and satisfies the functional equation A(k,z)^(k + 1) = 1/(1 - z)*( BINOMIAL(BINOMIAL(A(k,z))) )^k, where BINOMIAL(F(z))= 1/(1 - z)*F(z/(1 - z)) denotes the binomial transform of the o.g.f. F(z). A(k,z) = A(-(k + 1),-z). Cf. A019538.
For cases see A258377 (k = 1), A258378(k = 2), A258379 (k = 3), A258380 (k = 4) and A258381 (k = 5). (End)
T(n,k) = A154537(n,k)*k! = A039755(n,k)*(2^k*k!), 0 <= k <= n. - Wolfdieter Lang, Apr 19 2017
From Peter Bala, Jan 12 2018: (Start)
n-th row polynomial R(n,x) =  (1 + 2*x) o (1 + 2*x) o ... o (1 + 2*x) (n factors), where o denotes the black diamond multiplication operator of Dukes and White. See example E13 in the Bala link.
R(n,x) = Sum_{k = 0..n} binomial(n,k)*2^k*F(k,x) where F(k,x) is the Fubini polynomial of order k, the k-th row polynomial of A019538. (End)

A001249 Squares of tetrahedral numbers: a(n) = binomial(n+3,n)^2.

Original entry on oeis.org

1, 16, 100, 400, 1225, 3136, 7056, 14400, 27225, 48400, 81796, 132496, 207025, 313600, 462400, 665856, 938961, 1299600, 1768900, 2371600, 3136441, 4096576, 5290000, 6760000, 8555625, 10732176, 13351716, 16483600, 20205025, 24601600, 29767936, 35808256

Offset: 0

N. J. A. Sloane

Total area of all square and rectangular regions from an n+1 X n+1 grid. E.g., n = 2, there are 9 individual squares, 4 2 X 2's and 1 3 X 3, total area 9 + 16 + 9 = 34. The rectangular regions include 6 2 X 1's, 6 1 X 2's, 3 3 X 1's, 3 1 X 3's, 2 3 X 2's and 2 2 X 3's, total area 12 + 12 + 9 + 9 + 12 + 12 = 66, hence a(2) = 34 + 66 = 100. - Jon Perry, Jul 29 2003 [Index/grid size adjusted by Rick L. Shepherd, Jun 27 2017]
Number of 3 X 3 submatrices of an n+3 X n+3 matrix. - Rick L. Shepherd, Jun 27 2017
The inverse binomial transform gives row n=2 of A087107. - R. J. Mathar, Aug 31 2022

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000290, A000292, A006542, A033455, A108674 (first diffs.), A086020 (partial sums).
Third column of triangle A008459.
Cf. A000579, A001303, A040977.

A001249 := proc(n) binomial(n+3,n)^2 end proc: seq(A001249(n),n=0..10) ; # Zerinvary Lajos, May 17 2006

Table[Binomial[n + 3, 3]^2, {n, 0, 100}] (* T. D. Noe, Jun 26 2012 *)

a(n)=binomial(n+3,3)^2 \\ Charles R Greathouse IV, Sep 24 2015

From R. J. Mathar, Aug 19 2008: (Start)
a(n) = (A000292(n+1))^2.
O.g.f.: (1+x)(x^2+8x+1)/(1-x)^7. (End)
a(n) = C(n+4, 3)*C(n+4, 4)/(n+4) + A001303(n) = C(n+4, 3)*C(n+3, 3)/4 + A001303(n) = C(n+4, 6) + 3*C(n+5, 6) + C(n+6,6) + A001303(n). - Gary Detlefs, Aug 07 2013
-n^2*a(n) + (n+3)^2*a(n-1) = 0. - R. J. Mathar, Aug 15 2013
a(n) = 9*A040977(n-1) + A000579(n+6) + A000579(n+3). - R. J. Mathar, Aug 15 2013
a(n) = (n+3)*C(n+2, 2)*C(n+3, 3)/3. - Gary Detlefs, Jan 06 2014
a(n) = A000290(n+1)*A000290(n+2)*A000290(n+3)/36. - Bruno Berselli, Nov 12 2014
G.f. 2F1(4,4;1;x). - R. J. Mathar, Aug 09 2015
E.g.f.: exp(x)*(1 + 15*x + 69*x^2/2! + 147*x^3/3! + 162*x^4/4! +  90*x^5/5! +  20*x^6/6!). Computed from the o.g.f with the formulas (23) - (25) of the W. Lang link given in A060187. - Wolfdieter Lang, Jul 27 2017
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 9*Pi^2 - 351/4.
Sum_{n>=0} (-1)^n/a(n) = 63/4 - 3*Pi^2/2. (End)
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Wesley Ivan Hurt, Aug 29 2022
a(n) = a(n-1)+A000217(n+1)*A000330(n+1). - J. M. Bergot, Aug 29 2022
a(n) = A002415(n+2)^2 - 20*A006857(n-1). - Yasser Arath Chavez Reyes, Nov 08 2024

A051001 Sum of 4th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 82, 1, 626, 82, 2402, 1, 6643, 626, 14642, 82, 28562, 2402, 51332, 1, 83522, 6643, 130322, 626, 196964, 14642, 279842, 82, 391251, 28562, 538084, 2402, 707282, 51332, 923522, 1, 1200644, 83522, 1503652, 6643, 1874162, 130322, 2342084, 626, 2825762, 196964

Offset: 1

Eric W. Weisstein

Robert Israel, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher

Cf. A000265, A000593, A001227, A001159, A050999, A051000, A051002.

f:= proc(n) add(x^4, x = numtheory:-divisors(n/2^padic:-ordp(n,2))) end proc:
map(f, [$1..100]); # Robert Israel, Jan 05 2017

Table[Total[Select[Divisors[n],OddQ]^4],{n,40}] (* Harvey P. Dale, Oct 02 2014 *)
f[2, e_] := 1; f[p_, e_] := (p^(4*e + 4) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

a(n) = sumdiv(n , d, (d%2)*d^4); \\ Michel Marcus, Jan 14 2014

from sympy import divisor_sigma
def A051001(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),4)) # Chai Wah Wu, Jul 16 2022

Dirichlet g.f. (1-2^(4-s))*zeta(s)*zeta(s-4). - R. J. Mathar, Apr 06 2011
G.f.: Sum_{k>=1} (2*k - 1)^4*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Jan 04 2017
a(n) = A001159(A000265(n)). - Robert Israel, Jan 05 2017
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(4*e+4)-1)/(p^4-1) for p > 2. - Amiram Eldar, Sep 14 2020
Sum_{k=1..n} a(k) ~ zeta(5)*n^5/10. - Vaclav Kotesovec, Sep 24 2020
G.f.: Sum_{n >= 1} x^n*(1 + 76*x^(2*n) + 230*x^(4*n) + 76*x^(6*n) + x^(8*n))/(1 - x^(2*n))^5. See row 5 of A060187. - Peter Bala, Dec 20 2021

A002436 E.g.f.: Sum_{n >= 0} a(n)x^(2n)/(2n)! = sec(2x).

Original entry on oeis.org

1, 4, 80, 3904, 354560, 51733504, 11070525440, 3266330312704, 1270842139934720, 630424777638805504, 388362339077351014400, 290870261262635870715904, 260290690801376575335956480, 274278793184290987427604987904, 336150887870579862992197737512960

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 4*x + 80*x^2 + 3904*x^3 + 354560*x^4 + 51733504*x^5 + 11070525440*x^6 + ...

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 75.
J. W. L. Glaisher, On the last two figures in certain coefficients analogous to the Eulerian numbers, Quart. J. Pure Appl. Math., 44 (1913), 105-112.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..216

Cf. A000364, A000816, A000831.

(-1)^n*a(n) give the alternating row sums of A060187(2*n), n >= 0. The alternating sums for odd numbered rows vanish. - Wolfdieter Lang, Jul 12 2017

m:=35; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1+Tan(x))/(1-Tan(x)) )); [Factorial(n-1)*b[n]: n in [1..m by 2]]; // Vincenzo Librandi, May 30 2019

A := n -> (-4)^n*euler(2*n); # (Then A(n) = a(n+1) for n >= 0.) # Peter Luschny, Jan 27 2009

Rest@ Union[ Range[0, 24]! CoefficientList[ Series[ Sec[ 2x], {x, 0, 24}], x]] (* Robert G. Wilson v, Apr 16 2011 *)
a[ n_] :=  If[ n < 0, 0, 2 (-16)^n LerchPhi[ -1, -2 n, 1/2]]; (* Michael Somos, Oct 14 2014 *)
With[{nn=30},Take[CoefficientList[Series[Sec[2x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, May 06 2018 *)

{a(n) = local(m); if( n<0, 0, m = 2*n; m! * polcoeff( 1 / cos( 2*x + x * O(x^m)), m))}; /* Michael Somos, Apr 16 2011 */

@CachedFunction
def sp(n,x) :
    if n == 0 : return 1
    return -add(2^(n-k)*sp(k,1/2)*binomial(n,k) for k in range(n)[::2])
A002436 = lambda n : abs(sp(2*(n-1),x))
[A002436(n) for n in (1..15)]   # Peter Luschny, Jul 30 2012

a(n) = A000831(2*n) = 4^n * A000364(n). a(n) = 2 * A000816(n) except n=0. - Michael Somos, Apr 26 2011
E.g.f.: sec(2*x) = 1 + 2*(x^2)/G(0); G(k) = (k+1)*(2*k+1) - 2*(x^2) + (x^2)*(2*k+1)*(2*k+2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 01 2011
E.g.f.: sec(2*x) = 1/cos(2*x) = 1/(cos(x)^2 - sin(x)^2). - Arkadiusz Wesolowski, Jul 25 2012
From Sergei N. Gladkovskii, Oct 23 2012 (Start)
G.f.: 1/U(0) where U(k) = 1 - 2*(4*k+1)*(4*k+2)*x/(1 - 2*(4*k+3)*(4*k+4)*x/U(k+1)); (continued fraction, 2-step).
E.g.f.: 1/S(0) where S(k) = 1 - 2*x^2/((4*k+1)*(2*k+1) - x^2*(4*k+1)*(2*k+1)/(x^2 - (4*k+3)*(k+1)/S(k+1))); (continued fraction, 3rd kind, 3-step). (End)
G.f.: 1/U(0) where U(k) = 1 - (4*k+2)*(4*k+2)*x^2/(1 - (4*k+4)*(4*k+4)*x^2/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Nov 06 2012
G.f.: 1/G(0) where G(k) = 1 - 4*x*(k+1)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 12 2013
a(n+1) = | 2*16^n*lerchphi(-1, -2*n, 1/2) |, n>=0. - Peter Luschny, Apr 27 2013
G.f.: Q(0), where Q(k) = 1 - x*(2*k+2)^2/( x*(2*k+2)^2 - 1/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013
E.g.f.: sec(2*x) = 1/cos(2*x) = 1 + 2*x^2/(1-2*x^2)*T(0), where T(k) = 1 - x^2*(2*k+1)*(2*k+2)/( x^2*(2*k+1)*(2*k+2) + ((k+1)*(2*k+1) - 2*x^2)*((k+2)*(2*k+3) - 2*x^2)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013
a(n) = (-1)^n*2^(6*n+1)*(Zeta(-2*n,1/4) - Zeta(-2*n, 3/4)), where Zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015

More terms from Michael Somos, Jun 21 2002

A104035 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k>0 or if k<0; T(n,k) = kT(n-1,k-1) + (k+1)T(n-1,k+1).

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 5, 0, 6, 5, 0, 28, 0, 24, 0, 61, 0, 180, 0, 120, 61, 0, 662, 0, 1320, 0, 720, 0, 1385, 0, 7266, 0, 10920, 0, 5040, 1385, 0, 24568, 0, 83664, 0, 100800, 0, 40320, 0, 50521, 0, 408360, 0, 1023120, 0, 1028160, 0, 362880, 50521, 0, 1326122, 0, 6749040

Offset: 0

Philippe Deléham, Apr 06 2005

Or, triangle of coefficients (with exponents in increasing order) in polynomials Q_n(u) defined by d^n sec x / dx^n = Q_n(tan x)*sec x.
Interpolates between factorials and Euler (or secant) numbers. Related to Springer numbers.
Companion triangles are A155100 (derivative polynomials of tangent function) and A185896 (derivative polynomials of squared secant function).
A combinatorial interpretation for the polynomial Q_n(u) as the generating function for a sign change statistic on certain types of signed permutation can be found in [Verges]. A signed permutation is a sequence (x_1,x_2,...,x_n) of integers such that {|x_1|,|x_2|,...,|x_n|} = {1,2,...,n}. They form a group, the hyperoctahedral group of order 2^n*n! = A000165(n), isomorphic to the group of symmetries of the n dimensional cube.
Let x_1,...,x_n be a signed permutation. Adjoin x_0 = 0 to the front of the permutation and x_(n+1) = (-1)^n*(n+1) to the end to form x_0,x_1,...,x_n,x_(n+1). Then x_0,x_1,...,x_n,x_(n+1) is a snake of type S(n;0) when x_0 < x_1 > x_2 < ... x_(n+1). For example, 0 3 -1  2 -4 is a snake of type S(3;0).
Let sc be the number of sign changes through a snake ... sc = #{i, 0 <= i <= n, x_i*x_(i+1) < 0}. For example, the snake 0 3 -1 2 -4 has sc = 3. The polynomial Q_n(u) is the generating function for the sign change statistic on snakes of type S(n;0): ... Q_n(u) = sum {snakes in S(n;0)} u^sc. See the example section below for the cases n = 2 and n = 3.
PRODUCTION MATRIX
Let D = subdiag(1,2,3,...) be the array with the indicated sequence on the first subdiagonal and zeros elsewhere and let C = transpose(D). The production matrix for this triangle is C+D: the first row of (C+D)^n is the n-th row of this triangle. D represents the derivative operator d/dx and C represents the operator p(x) -> x*d/dx(x*p(x)) acting on the basis monomials {x^n}n>=0. See Formula (1) below.

			The polynomials Q_0(u) through Q_6(u) (with exponents in decreasing order) are:
  1
  u
  2*u^2 + 1
  6*u^3 + 5*u
  24*u^4 + 28*u^2 + 5
  120*u^5 + 180*u^3 + 61*u
  720*u^6 + 1320*u^4 + 662*u^2 + 61
Triangle begins:
  1
  0 1
  1 0 2
  0 5 0 6
  5 0 28 0 24
  0 61 0 180 0 120
  61 0 662 0 1320 0 720
  0 1385 0 7266 0 10920 0 5040
  1385 0 24568 0 83664 0 100800 0 40320
  0 50521 0 408360 0 1023120 0 1028160 0 362880
  50521 0 1326122 0 6749040 0 13335840 0 11491200 0 3628800
  0 2702765 0 30974526 0 113760240 0 185280480 0 139708800 0 39916800
  2702765 0 98329108 0 692699304 0 1979524800 0 2739623040 0 1836172800 0 479001600
Examples of sign change statistic sc on snakes of type S(n;0)
= = = = = = = = = = = = = = = = = = = = = =
.....Snakes....# sign changes sc.......u^sc
= = = = = = = = = = = = = = = = = = = = = =
n=2
...0 1 -2 3...........2.................u^2
...0 2  1 3...........0.................1
...0 2 -1 3...........2.................u^2
yields Q_2(u) = 2*u^2 + 1.
n=3
...0 1 -2  3 -4.......3.................u^3
...0 1 -3  2 -4.......3.................u^3
...0 1 -3 -2 -4.......1.................u
...0 2  1  3 -4.......1.................u
...0 2 -1  3 -4.......3.................u^3
...0 2 -3  1 -4.......3.................u^3
...0 2 -3 -2 -4.......1.................u
...0 3  1  2 -4.......1.................u
...0 3 -1  2 -4.......3.................u^3
...0 3 -2  1 -4.......3.................u^3
...0 3 -2 -1 -4.......1.................u
yields Q_3(u) = 6*u^3 + 5*u.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 287.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see pp. 445 and 469.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
K. Boyadzhiev, Derivative Polynomials for tanh, tan, sech and sec in Explicit Form, arXiv:0903.0117 [math.CA], 2009-2010.
M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005.
Gordon Haigh, A "natural" approach to Pick's theorem, Math. Gaz. 64 (1980), no. 429, 173-180.
Michael E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly, 102 (1995), 23-30.
Michael E. Hoffman, Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences, The Electronic Journal of Combinatorics, Volume 6.1 (1999): Research paper R21, 13 p.
M. Josuat-Vergès, Enumeration of snakes and cycle-alternating permutations, arXiv:1011.0929 [math.CO], 2010.
Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688.

See A008294 for another version of this triangle.
Setting u=0,1,2,3,4 gives A000364, A001586, A156129, A156131, A156132.
Setting u=sqrt(2) gives A156134 and A156138; u=sqrt(3) gives A002437 and A002439.
Cf. A060187, A155100, A185896, A186492.

a104035 n k = a104035_tabl !! n !! k
a104035_row n = a104035_tabl !! n
a104035_tabl = iterate f [1] where
   f xs = zipWith (+)
     (zipWith (*) [1..] (tail xs) ++ [0,0]) ([0] ++ zipWith (*) [1..] xs)
-- Reinhard Zumkeller, Apr 27 2014

nmax = 10; t[n_, k_] := t[n, k] = k*t[n-1, k-1] + (k+1)*t[n-1, k+1]; t[0, 0] = 1; t[0, ] = 0; Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 0, n}]] (* _Jean-François Alcover, Nov 14 2011 *)

T(n, n) = n!; T(n, 0) = 0 if n = 2m + 1; T(n, 0) = A000364(m) if n = 2m.
Sum_{k>=0} T(m, k)*T(n, k) = T(m+n, 0).
Sum_{k>=0} T(n, k) = A001586(n): Springer numbers.
G.f.: Sum_{n >= 0} Q_n(u)*t^n/n! = 1/(cos t - u sin t).
From Peter Bala: (Start)
RECURRENCE RELATION
For n>=0,
(1)... Q_(n+1)(u) = d/du Q_n(u) + u*d/du(u*Q_n(u))
... = (1+u^2)*d/du Q_n(u) + u*Q_n(u),
with starting condition Q_0(u) = 1. Compare with Formula (4) of A186492.
RELATION WITH TYPE B EULERIAN NUMBERS
(2)... Q_n(u) = ((u+i)/2)^n*B(n,(u-i)/(u+i)), where i = sqrt(-1) and
[B(n,u)]n>=0 = [1,1+u,1+6*u+u^2,1+23*u+23*u^2+u^3,...] is the sequence of type B Eulerian polynomials (with a factor of u removed) - see A060187.
(End)
T(n,0) = abs(A122045(n)). - Reinhard Zumkeller, Apr 27 2014

Entry revised by N. J. A. Sloane, Nov 06 2009

A051002 Sum of 5th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 244, 1, 3126, 244, 16808, 1, 59293, 3126, 161052, 244, 371294, 16808, 762744, 1, 1419858, 59293, 2476100, 3126, 4101152, 161052, 6436344, 244, 9768751, 371294, 14408200, 16808, 20511150, 762744, 28629152, 1, 39296688, 1419858, 52541808, 59293, 69343958

Offset: 1

Eric W. Weisstein

The Apostol exercise F(x) is the g.f. of a(n)*(-1)^(n+1). - Michael Somos, Jul 05 2021

			G.f. = x + x^2 + 244*x^3 + x^4 + 3126*x^5 + 244*x^6 + 16808*x^7 + x^8 + ... - _Michael Somos_, Jul 05 2021

T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25, Exercise 15.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher

Cf. A000593, A001227, A050999, A051000, A051001, A001160.

a[n_] := Select[ Divisors[n], OddQ]^5 // Total; Table[a[n], {n, 1, 34}] (* Jean-François Alcover, Oct 25 2012 *)
f[2, e_] := 1; f[p_, e_] := (p^(5*e + 5) - 1)/(p^5 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)
a[ n_] := If[n == 0, 0, DivisorSigma[5, n / 2^IntegerExponent[n, 2]]]; (* Michael Somos, Jul 05 2021 *)

a(n) = sumdiv(n , d, (d%2)*d^5); \\ Michel Marcus, Jan 14 2014

a(n)=sumdiv(n>>valuation(n,2), d, d^5) \\ Charles R Greathouse IV, Jul 05 2021

from sympy import divisor_sigma
def A051002(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),5)) # Chai Wah Wu, Jul 16 2022

Dirichlet g.f.: (1-2^(5-s))*zeta(s)*zeta(s-5). - R. J. Mathar, Apr 06 2011
G.f.: Sum_{k>=1} (2*k - 1)^5*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Jan 04 2017
The preceding g.f. is also 34*sigma_5(x^2) - 64*sigma_5(x^4) - sigma_5(-x), with sigma_5 the g.f. of A001160. Compare this with the Apostol reference which gives the g.f. of a(n)*(-1)^(n+1). - Wolfdieter Lang, Jan 31 2017
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(5*e+5)-1)/(p^5-1) for p > 2. - Amiram Eldar, Sep 14 2020
Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / 11340. - Vaclav Kotesovec, Sep 24 2020
G.f.: Sum_{n >= 1} x^n*R(5,x^(2*n))/(1 - x^(2*n))^6, where R(5,x) = 1 + 237*x + 1682*x^2 + 1682*x^3 + 237*x^4 + x^5 is the fifth row polynomial of A060187. - Peter Bala, Dec 20 2021

A174266 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(3n+1,i) binomial(k+3-i,3)^n, 0 <= k <= 3*(n-1).

Original entry on oeis.org

1, 1, 9, 9, 1, 1, 54, 405, 760, 405, 54, 1, 1, 243, 6750, 49682, 128124, 128124, 49682, 6750, 243, 1, 1, 1008, 83736, 1722320, 12750255, 40241088, 58571184, 40241088, 12750255, 1722320, 83736, 1008, 1, 1, 4077, 922347, 45699447, 789300477, 5904797049, 21475242671, 40396577931, 40396577931, 21475242671, 5904797049, 789300477, 45699447, 922347, 4077, 1

Offset: 1

Roger L. Bagula, Mar 14 2010

From Yahia Kahloune, Jan 30 2014: (Start)
In general, let b(k,e,p) = Sum_{i=0..k} (-1)^i*binomial(e*p+1,i)*binomial(k+e-i,e)^p. Then T(n,k) = b(k,3,n).
With these coefficients we can calculate: Sum_{i=1..n} binomial(i+e-1,e)^p = Sum_{i=0..e*(p-1)} b(i,e,p)*binomial(n+e+i,e*p+1).
For example, A086020(n) = Sum_{i=1..n} binomial(2+i, 3)^2 = T(2,0)*binomial(n+3, 7) + T(2,1)*binomial(n+4,7) + T(2,2)*binomial(n+5,7) + T(2,3)*binomial(n+6,7) = (1/5040)*(20*n^7 + 210*n^6 + 854*n^5 + 1680*n^4 + 1610*n^3 + 630*n^2 + 36*n). (End)
T(n,k) is the number of permutations of 3 indistinguishable copies of 1..n with exactly k descents. A descent is a pair of adjacent elements with the second element less than the first. - Andrew Howroyd, May 06 2020

			Triangle begins:
  1;
  1,      9,         9,            1;
  1,     54,       405,          760,            405,       54,        1;
  1,    243,      6750,        49682,         128124,   128124,    49682, ... ;
  1,   1008,     83736,      1722320,       12750255, 40241088, 58571184, ... ;
  1,   4077,    922347,     45699447,      789300477, ... ;
  1,  16362,   9639783,   1063783164,    38464072830, ... ;
  1,  65511,  98361900,  23119658500,  1641724670475, ... ;
  1, 262116, 992660346, 484099087156, 64856779908606, ... ;
...
The T(2,1) = 9 permutations of 111222 with 1 descent are: 112221, 112212, 112122, 122211, 122112, 121122, 222111, 221112, 211122. - _Andrew Howroyd_, May 07 2020

Andrew Howroyd, Table of n, a(n) for n = 1..1335
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 12.

Columns k=0..9 are A000012, A289254, A151632, A151633, A151634, A151635, A151636, A151637, A151638, A151639.
Row sums are A014606.
Similar triangles for e=1..6: A173018 (or A008292), A154283, this sequence, A236463, A237202, A237252.
Cf. A060187, A174264.

(* First program *)
p[n_, x_]:= p[n,x]= (1-x)^(3*n+1)*Sum[(Binomial[k+1, 3])^n*x^k, {k, 0, Infinity}]/x^2;
Table[CoefficientList[p[x, n], x], {n,10}]//Flatten (* corrected by G. C. Greubel, Mar 26 2022 *)
(* Second program *)
T[n_, k_]:= T[n, k]= Sum[(-1)^(k-j+1)*Binomial[3*n+1, k-j+1]*(j*(j^2-1)/2)^n, {j, 0, k+1}]/(3^n);
Table[T[n, k], {n,10}, {k,3*n-2}]//Flatten (* G. C. Greubel, Mar 26 2022 *)

T(n,k)={sum(i=0, k, (-1)^i*binomial(3*n+1, i)*binomial(k+3-i, 3)^n)} \\ Andrew Howroyd, May 06 2020

@CachedFunction
def T(n, k): return (1/3^n)*sum( (-1)^(k-j+1)*binomial(3*n+1, k-j+1)*(j*(j^2-1)/2)^n for j in (0..k+1) )
flatten([[T(n, k) for k in (1..3*n-2)] for n in (1..10)]) # G. C. Greubel, Mar 26 2022

T(n,k) = [x^k] (1-x)^(3*n+1)*(Sum_{k>=0} (k*(k+1)*(k-1)/2)^n*x^k)/(3^n*x^2).
T(n,k) = T(n, 3*n-k).
From Yahia Kahloune, Jan 30 2014: (Start)
Sum_{i=1..n} binomial(2+i,3)^p = Sum_{i=0..3*p-3} T(p,i)*binomial(n+3+i,3*p+1).
binomial(n,3)^p = Sum_{i=0..3*p-3} T(p,i)*binomial(n+i,3*p). (End)
From Sergii Voloshyn, Dec 18 2024: (Start)
Let E be the operator (x^2)D*(1/x)*D*(x^2)*D, where D denotes the derivative operator d/dx. Then (1/6^n)* E^n(x^2/(1 - x)^4) = (row n generating polynomial)/(1 - x)^(3*n+4) = Sum_{i=0..k} (-1)^i * binomial(3*n+1,i) * binomial(k+3-i,3)^n.
For example, when n = 3 we have 1/216*E^3(x^2/(1 - x)^4) = x^2 (1 + 243x + 6750x^2 + 49682x^3 + 128124x^4 + 128124x^5 + 49682x^6 + 6750x^7 + 243x^8 + x^9)/(1 - x)^13. (End)

Edited by Andrew Howroyd, May 06 2020

A154537 Triangle T(n,m) read by rows: let p(n,x) = exp(-x) * Sum_{m >= 0} (2m + 1)^n x^m/m!; then T(n,m) = [x^m] p(n,x).

Original entry on oeis.org

1, 1, 2, 1, 8, 4, 1, 26, 36, 8, 1, 80, 232, 128, 16, 1, 242, 1320, 1360, 400, 32, 1, 728, 7084, 12160, 6320, 1152, 64, 1, 2186, 36876, 99288, 81200, 25312, 3136, 128, 1, 6560, 188752, 768768, 929376, 440832, 91392, 8192, 256, 1, 19682, 956880, 5758880, 9901920, 6707904, 2069760, 305664, 20736, 512

Offset: 0

Roger L. Bagula, Jan 11 2009

Row sums are A126390.
These numbers are related to Stirling numbers of the second kind as MacMahon numbers A060187 are related to Eulerian numbers.
Let p and q denote operators acting on a function f(x) by pf(x) = x*f(x) and qf(x) = d/dx(f(x)). Let A be the anticommutator operator qp + pq. Then A^n = Sum_{k = 0..n} T(n,k) p^k q^k. For example, A^3(f) = f + 26*x*df/dx + 36*x^2*d^2(f)/dx^2 + 8*x^3*d^3(f)/dx^3. - Peter Bala, Jul 24 2014
From Peter Bala, May 21 2023: (Start)
Compare the definition of the polynomial p(n,x) with Dobiński's formula for the Bell polynomials (row polynomials of A008277 for n >= 1): Bell(n,x) = exp(-x) * Sum_{m >= 0} m^n * x^m/m!.
Boyadzhiev has shown that Bell(n,x) = d/dx( exp(-x) * Sum_{m >= 0} (1^n + 2^n + ... + (m-1)^n) * x^m/m! ). The corresponding result for this table is that the n-th row polynomial p(n,x) = d/dx( exp(-x) * Sum_{m >= 0} (1^n + 3^n + ... + (2*m-1)^n) * x^m/m! ). (End)

			Triangle begins:
  {1},
  {1, 2},
  {1, 8, 4},
  {1, 26, 36, 8},
  {1, 80, 232, 128, 16},
  {1, 242, 1320, 1360, 400, 32},
  {1, 728, 7084, 12160, 6320, 1152, 64},
  {1, 2186, 36876, 99288, 81200, 25312, 3136, 128},
  {1, 6560, 188752, 768768, 929376, 440832, 91392, 8192, 256},
  {1, 19682, 956880, 5758880, 9901920, 6707904, 2069760, 305664, 20736, 512},
  ...
Boas-Buck recurrence for column m = 2, and n = 4: T(4,2) = (1/2)*[4*3*T(3, 2) + 2*6*(-2)^2*Bernoulli(2)*T(2,2)] = (1/2)*(12*36 + 12*4*(1/6)*4) = 232. - _Wolfdieter Lang_, Aug 11 2017

Khristo N. Boyadzhiev, New identities with Stirling, hyperharmonic, and derangement numbers, Bernoulli and Euler polynomials, powers, and factorials, arXiv:2011.03101v3 [math.NT], 2020-2021.
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 9.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Eric Weisstein's World of Mathematics, Dobiński's formula

Cf. A008277, A000110, A039755, A048993, A060187.

p[x_, n_] = Sum[(2*m + 1)^n*x^m/m!, {m, 0, Infinity}]/(Exp[x]);
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}]
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]

From Peter Bala, Oct 28 2011: (Start)
T(n,k) = 1/k!*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(2*j+1)^n.
Recurrence relation: T(n,k) = 2*T(n-1,k-1) + (2*k+1)*T(n-1,k).
T(n,k) = (2^k)*A039755(n,k).
E.g.f.: exp(x + y*(exp(2*x) - 1)) = 1 + (1 + 2*y)*x + (1 + 8*y + 4*y^2)*x^2/2! + .... (End)
T(n, k) = Sum_{m=0..n} binomial(n, m)*2^m*Stirling2(m, k), 0 <= k <= n, where Stirling2 is A048993. - Wolfdieter Lang, Apr 13 2017
Boas-Buck recurrence for column sequence m: T(n,k) = (1/(n - k))*[n*(1 + m)*T(n-1,k) + k*Sum_{p=m..n-2} binomial(n,p)*(-2)^(n-p)*Bernoulli(n-p)*T(p,k)], for n > m >= 0, with input T(m,m) = 2^m. See a comment in A282629, also for references, and an example below. - Wolfdieter Lang, Aug 11 2017

Edited by N. J. A. Sloane, Jan 12 2009

A257609 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = 2*x + 2.

Original entry on oeis.org

1, 2, 2, 4, 16, 4, 8, 88, 88, 8, 16, 416, 1056, 416, 16, 32, 1824, 9664, 9664, 1824, 32, 64, 7680, 76224, 154624, 76224, 7680, 64, 128, 31616, 549504, 1999232, 1999232, 549504, 31616, 128, 256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 256

Offset: 0

Dale Gerdemann, May 03 2015

			Triangle begins as:
    1;
    2,      2;
    4,     16,       4;
    8,     88,      88,        8;
   16,    416,    1056,      416,       16;
   32,   1824,    9664,     9664,     1824,       32;
   64,   7680,   76224,   154624,    76224,     7680,      64;
  128,  31616,  549504,  1999232,  1999232,   549504,   31616,    128;
  256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 256;

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

Cf. A000079, A002866 (row sums), A008292, A060187, A100575, A257611, A257613.
Cf. A257615, A038208, A256890, A257610, A257612, A257614, A257616, A257617.
Cf. A257618, A257619.
Similar sequences listed in A256890.

function T(n,k,a,b)
  if k lt 0 or k gt n then return 0;
  elif k eq 0 or k eq n then return 1;
  else return (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b);
  end if; return T;
end function;
[T(n,k,2,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 21 2022

A257609:= func< n,k | 2^n*EulerianNumber(n+1, k) >;
[A257609(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 17 2025

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,2,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)

def T(n,k,a,b): # A257609
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,2,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.
Sum_{k=0..n} T(n, k) = A002866(n).
From G. C. Greubel, Mar 21 2022: (Start)
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 2, and b = 2.
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
T(n, 1) = 2*A100575(n+1). (End)
T(n, k) = 2^n*A008292(n+1, k+1). - G. C. Greubel, Jan 17 2025

cp's OEIS Frontend

A144431 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n,1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = -1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A145901 Triangle of f-vectors of the simplicial complexes dual to the permutohedra of type B_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A001249 Squares of tetrahedral numbers: a(n) = binomial(n+3,n)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A051001 Sum of 4th powers of odd divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A002436 E.g.f.: Sum_{n >= 0} a(n)*x^(2*n)/(2*n)! = sec(2*x).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A104035 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k>0 or if k<0; T(n,k) = k*T(n-1,k-1) + (k+1)*T(n-1,k+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A144431 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n,1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = -1.

A002436 E.g.f.: Sum_{n >= 0} a(n)x^(2n)/(2n)! = sec(2x).

A104035 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k>0 or if k<0; T(n,k) = kT(n-1,k-1) + (k+1)T(n-1,k+1).

A174266 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(3n+1,i) binomial(k+3-i,3)^n, 0 <= k <= 3*(n-1).

A154537 Triangle T(n,m) read by rows: let p(n,x) = exp(-x) * Sum_{m >= 0} (2m + 1)^n x^m/m!; then T(n,m) = [x^m] p(n,x).

A257609 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = 2*x + 2.