A005990 -id:A005990 - cp's OEIS frontend

A062193 Fourth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).

1, 24, 420, 6720, 105840, 1693440, 27941760, 479001600, 8562153600, 159826867200, 3116623910400, 63465795993600, 1348648164864000, 29877743960064000, 689322235650048000, 16543733655601152000, 412559358036553728000, 10678006913887272960000, 286526518855975157760000

Offset: 0

Wolfdieter Lang, Jun 19 2001

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for sequences related to Laguerre polynomials.

Cf. A001710, A005990, A005461, A062139.

Cf. A001113, A001620, A091725, A099285.

[Factorial(n+3)*binomial(n+5, 5)/Factorial(3): n in [0..30]]; // G. C. Greubel, May 11 2018

With[{nn=20},CoefficientList[Series[(1+15*x+30*x^2+10*x^3)/(1-x)^9, {x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 02 2018 *)

{ f=2; for (n=0, 100, f*=n + 3; write("b062193.txt", n, " ", f*binomial(n + 5, 5)/6) ) } \\ Harry J. Smith, Aug 02 2009

my(x='x+O('x^30)); Vec(serlaplace((1+15*x+30*x^2+10*x^3)/(1-x)^9)) \\ G. C. Greubel, May 11 2018

[binomial(n,5)*factorial (n-2)/6 for n in range(5, 21)] # Zerinvary Lajos, Jul 07 2009

E.g.f.: (1+15*x+30*x^2+10*x^3)/(1-x)^9.
a(n) = A062139(n+3, 3).
a(n) = (n+3)!*binomial(n+5, 5)/3!.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j) then a(n-3) = (-1)^(n-1)*f(n,3,-6), (n>=3). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 75*(Ei(1) - gamma) - 30*e - 65/4, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=0} (-1)^n/a(n) = 315*(gamma - Ei(-1)) - 180/e - 735/4, where Ei(-1) = -A099285. (End)

A324224 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 6, 4, 1, 1, 6, 18, 24, 18, 6, 1, 1, 8, 36, 96, 120, 96, 36, 8, 1, 1, 10, 60, 240, 600, 720, 600, 240, 60, 10, 1, 1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 35280, 15120, 4200, 840, 126, 14, 1

Offset: 1

Alois P. Heinz, Feb 18 2019

			Triangle T(n,k) begins:
  :                                 1                              ;
  :                           1,    2,    1                        ;
  :                     1,    4,    6,    4,    1                  ;
  :               1,    6,   18,   24,   18,    6,   1             ;
  :          1,   8,   36,   96,  120,   96,   36,   8,  1         ;
  :      1, 10,  60,  240,  600,  720,  600,  240,  60, 10,  1     ;
  :  1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1  ;

Alois P. Heinz, Rows n = 1..100, flattened
Wikipedia, Permutation
Wikipedia, Permutation matrix

Columns k=0-6 give (offsets may differ): A000142, A001563, A001286, A005990, A061206, A062199, A062148.
Row sums give A306495(n-1).
Cf. A132159 (right part of triangle), A306234, A324225.

b:= proc(s, c) option remember; (n-> `if`(n=0, c,
      add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t!*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t

T[n_, k_] := With[{t = Abs[k]}, If[tJean-François Alcover, Mar 25 2021, after 3rd Maple program *)

T(n,k) = T(n,-k).
T(n,k) = (n-t)*(n-1)!/t! if t < n with t = |k|, T(n,k) = 0 otherwise.
T(n,k) = 1/|k|! * A324225(n,k).
E.g.f. of column k: x^t/t! * hypergeom([2, t], [t+1], x) with t = |k|+1.
Sum_{k=1-n..n-1} T(n,k) = A306495(n-1).

A062194 Fifth column sequence of triangle A062139 (generalized a=2 Laguerre).

Original entry on oeis.org

1, 35, 840, 17640, 352800, 6985440, 139708800, 2854051200, 59935075200, 1298593296000, 29088489830400, 674324082432000, 16183777978368000, 402104637462528000, 10339833534750720000, 275039572024369152000

Offset: 0

Wolfdieter Lang, Jun 19 2001

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for sequences related to Laguerre polynomials.

Cf. A001710, A005461, A005990, A062193.

Cf. A001113, A001620, A091725, A099285.

List([0..15],n->Factorial(n+4)*Binomial(n+6,6)/Factorial(4)); # Muniru A Asiru, Jul 01 2018

[Factorial(n+4)*Binomial(n+6, 6)/Factorial(4): n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+4)!*Binomial[n+6,6]/4!, {n, 0, 20}] (* G. C. Greubel, May 12 2018 *)

{ f=6; for (n=0, 100, f*=n + 4; write("b062194.txt", n, " ", f*binomial(n + 6, 6)/24) ) } \\ Harry J. Smith, Aug 02 2009

[binomial(n,6)*factorial (n-2)/factorial (4) for n in range(6, 22)] # Zerinvary Lajos, Jul 07 2009

E.g.f.: (1 + 24*x + 90*x^2 + 80*x^3 + 15*x^4)/(1-x)^11.
a(n) = A062139(n+4, 4).
a(n) = (n+4)!*binomial(n+6, 6)/4!.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..n} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n-4) = (-1)^n*f(n,4,-7), (n >= 4). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 336*(gamma - Ei(1)) - 96*e + 3524/5, where gamma = A001620, Ei(1) = A091725, and e = A001113.
Sum_{n>=0} (-1)^n/a(n) = 3264*(gamma - Ei(-1)) - 1920/e - 9464/5, where Ei(-1) = -A099285. (End)

A138770 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} such that there are exactly k entries between the entries 1 and 2 (n>=2, 0<=k<=n-2).

Original entry on oeis.org

2, 4, 2, 12, 8, 4, 48, 36, 24, 12, 240, 192, 144, 96, 48, 1440, 1200, 960, 720, 480, 240, 10080, 8640, 7200, 5760, 4320, 2880, 1440, 80640, 70560, 60480, 50400, 40320, 30240, 20160, 10080, 725760, 645120, 564480, 483840, 403200, 322560, 241920, 161280, 80640

Offset: 2

Emeric Deutsch, Apr 06 2008

Sum of row n = n! = A000142(n).

The expected value of k is (n-2)/3. [Geoffrey Critzer, Dec 19 2009]

			T(4,2)=4 because we have 1342, 1432, 2341 and 2431.
Triangle starts:
  2;
  4,2;
  12,8,4;
  48,36,24,12;
  240,192,144,96,48;
  ...

Cf. A000142, A052489, A052582, A052609, A005990, A061206, A052849, A090672.

T:=proc(n,k) if n-2 < k then 0 else (2*n-2*k-2)*factorial(n-2) end if end proc; for n from 2 to 10 do seq(T(n, k),k=0..n-2) end do; # yields sequence in triangular form

Table[Table[2 (n - r) (n - 2)!, {r, 1, n - 1}], {n, 1, 10}] // Grid (* Geoffrey Critzer, Dec 19 2009 *)

T(n,k) = 2*(n-k-1)*(n-2)!.
T(n,0) = 2(n-1)! = A052849(n-1).
T(n,1) = A052582(n-2).
T(n,2) = A052609(n-2).
T(n,3) = 12*A005990(n-3).
T(n,4) = 48*A061206(n-5).
T(n,n-2) = 2(n-2)! (A052849).
Sum_{k=0..n-2} k*T(n,k) = n!*(n-2)/3 = A090672(n-1).

A167568 A triangle related to the GF(z) formulas of the rows of the ED2 array A167560.

Original entry on oeis.org

1, 0, 2, 2, -2, 6, 0, 16, -16, 24, 24, -48, 144, -120, 120, 0, 432, -864, 1392, -960, 720, 720, -2160, 8208, -12816, 14448, -8400, 5040, 0, 23040, -69120, 149760, -184320, 161280, -80640, 40320, 40320, -161280, 760320, -1716480, 2684160, -2695680

Offset: 1

Johannes W. Meijer, Nov 10 2009

The GF(z) formulas given below correspond to the first ten rows of the ED2 array A167560. The polynomials in their numerators lead to the triangle given above.

			Row 1: GF(z) = 1/(1-z).
Row 2: GF(z) = 2/(1-z)^2.
Row 3: GF(z) = (2*z^2 - 2*z + 6)/(1-z)^3.
Row 4: GF(z) = (0*z^3 + 16*z^2 - 16*z + 24)/(1-z)^4.
Row 5: GF(z) = (24*z^4 - 48*z^3 + 144*z^2 - 120*z + 120)/(1-z)^5.
Row 6: GF(z) = (432*z^4 - 864*z^3 + 1392*z^2 - 960*z + 720)/(1-z)^6.
Row 7: GF(z) = (720*z^6 - 2160*z^5 + 8208*z^4 - 12816*z^3 + 14448*z^2 - 8400*z + 5040)/(1-z)^7.
Row 8: GF(z) = (0*z^7 + 23040*z^6 - 69120*z^5 + 149760*z^4 - 184320*z^3 + 161280*z^2 - 80640*z + 40320)/(1-z)^8.
Row 9: GF(z) = (40320*z^8 - 161280*z^7 + 760320*z^6 - 1716480*z^5 + 2684160*z^4 - 2695680*z^3 + 1935360*z^2 - 846720*z + 362880)/(1-z)^9.
Row 10: GF(z) = (0*z^9 + 2016000*z^8 - 8064000*z^7 + 22464000*z^6 - 39168000*z^5 + 48360960*z^4 - 40849920*z^3 + 24917760*z^2 - 9676800*z + 3628800)/(1-z)^10.

A167560 is the ED2 array.
A005359 equals the first left hand column.
A000142(n=>1) and 2*A005990 equal the first two right hand columns.
A000142(n=>1) equals the row sums.

A317527 Number of edges in the n-alternating group graph.

Original entry on oeis.org

0, 0, 3, 24, 180, 1440, 12600, 120960, 1270080, 14515200, 179625600, 2395008000, 34248614400, 523069747200, 8499883392000, 146459529216000, 2667655710720000, 51218989645824000, 1033983353475072000, 21896118073589760000, 485363950631239680000, 11240007277776076800000

Offset: 1

Eric W. Weisstein, Jul 30 2018

nonn

Looks like the denominators of g.f.: (1-x)*exp(-x) + x^2*(Chi(x) - Shi(x)), for cosh and sinh integral functions. - Benedict W. J. Irwin, Jun 04 2018

Eric Weisstein's World of Mathematics, Alternating Group Graph
Eric Weisstein's World of Mathematics, Edge Count

Cf. A005990, A181967.

[0] cat [Factorial(n)*(n-2)/2: n in [2..25]]; // Vincenzo Librandi, Jul 31 2018

Join[{0}, Table[n! (n - 2)/2, {n, 2, 20}]]
CoefficientList[Series[x^2/(2 (-1 + x)^2), {x, 0, 19}], x] Range[20]!

a(n) = n!*(n - 2)/2 for n > 1.
a(n) = 3 * A005990(n-1) for n>1. - Alois P. Heinz, Jul 30 2018
E.g.f.: x^3/(2*(x - 1)^2).

A062195 Sixth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).

Original entry on oeis.org

1, 48, 1512, 40320, 997920, 23950080, 570810240, 13699445760, 333923990400, 8310997094400, 211930425907200, 5548723878297600, 149353151057510400, 4135933413900288000, 117874102296158208000

Offset: 0

Wolfdieter Lang, Jun 19 2001

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for sequences related to Laguerre polynomials.

Cf. A001710, A005990, A005461, A062139, A062193, A062194.

Cf. A001113, A001620, A091725, A099285.

[Factorial(n+5)*Binomial(n+7, 7)/Factorial(5): n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+5)!*Binomial[n+7, 7]/5!, {n, 0, 20}] (* G. C. Greubel, May 12 2018 *)

{ f=24; for (n=0, 100, f*=n + 5; write("b062195.txt", n, " ", f*binomial(n + 7, 7)/120) ) } \\ Harry J. Smith, Aug 02 2009

E.g.f.: N(2;5, x)/(1-x)^13 with N(2;5, x) := Sum_{k=0..5} A062196(5, k)*x^k = 1+35*x+210*x^2+350*x^3+175*x^4+21*x^5.
a(n) = A062139(n+5, 5).
a(n) = (n+5)!*binomial(n+7, 7)/5!.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n-5) = (-1)^(n-1)*f(n,5,-8), (n>=5). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 1295*(Ei(1) - gamma) + 2170*e - 22813/3, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=0} (-1)^n/a(n) = 36575*(gamma - Ei(-1)) - 21700/e - 63455/3, where Ei(-1) = -A099285. (End)

A193474 Table read by rows: The coefficients of the polynomials P(n, x) = Sum{k=0..n} Sum{j=0..k} (-1)^j * 2^(-k) * binomial(k, j) * (k-2j)^n x^(n-k).

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 6, 0, 1, 0, 24, 0, 8, 0, 0, 120, 0, 60, 0, 1, 0, 720, 0, 480, 0, 32, 0, 0, 5040, 0, 4200, 0, 546, 0, 1, 0, 40320, 0, 40320, 0, 8064, 0, 128, 0, 0, 362880, 0, 423360, 0, 115920, 0, 4920, 0, 1, 0, 3628800, 0, 4838400, 0, 1693440, 0, 130560, 0, 512, 0, 0

Offset: 1

Peter Luschny, Aug 01 2011

See A196776 for a row reversed form of this triangle. - Peter Bala, Oct 06 2011

			The sequence of polynomials P(n, x) begins:
[0]    1;
[1]    1;
[2]    2;
[3]    6 +      x^2;
[4]   24 +    8*x^2;
[5]  120 +   60*x^2 +     x^4;
[6]  720 +  480*x^2 +  32*x^4;
[7] 5040 + 4200*x^2 + 546*x^4 + x^6.

Cf. A196776, A000142, A006154, A191277, A000111, A000828, A000557, A107403, A007289, A005990.

A193474_polynom := proc(n,x) local k, j;
add(add((-1)^j*2^(-k)*binomial(k,j)*(k-2*j)^n*x^(n-k),j=0..k),k=0..n) end: seq(seq(coeff(A193474_polynom(n,x),x,i),i=0..n),n=0..10);

p[n_, x_] := Sum[(-1)^j*2^(-k)*Binomial[k, j]*(k-2*j)^n*x^(n-k), {k, 0, n}, {j, 0, k}]; t[n_, k_] := Coefficient[p[n, x], x, k]; t[0, 0] = 1; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2014 *)

P(n, 0) = A000142(n).
P(n, 1) = A006154(n).
P(n, 2) = A191277(n).
P(n, i) = A000111(n+1), where i is the imaginary unit.
P(n, i)*2^n = A000828(n+1).
P(n, 1/2)*2^n = A000557(n).
P(n, 1/3)*3^n = A107403(n).
P(n, i/2)*2^n = A007289(n).
G(m, x) = 1/(1 - m*sinh(x)) is the generating function of m^n*P(n, 1/m).
GI(m, x) = 1/(1 - m*sin(x)) is the generating function of m^n*P(n, i/m).
[x^2] P(n+1, x) = A005990(n).

A166553 Triangle read by rows: T(n, k) = [x^k]( (n+2)!(3EulerE(n, x+1) - EulerE(n, x))/4 ).

Original entry on oeis.org

1, 3, 3, 0, 24, 12, -30, 0, 180, 60, 0, -720, 0, 1440, 360, 2520, 0, -12600, 0, 12600, 2520, 0, 120960, 0, -201600, 0, 120960, 20160, -771120, 0, 3810240, 0, -3175200, 0, 1270080, 181440, 0, -61689600, 0, 101606400, 0, -50803200, 0, 14515200, 1814400

Offset: 0

Roger L. Bagula, Dec 12 2010

I think the rows are indexed by t = 0, 1, 2, ..., and in each row we expand the polynomial in powers of x. - N. J. A. Sloane, Dec 14 2010

Former name: Triangle read by rows: expansion of p(x,t) = exp(x*t)*(3*exp(t) - 1)/(exp(t) + 1), with coefficient of x^n scaled by multiplication by (n!*(n + 2)!/4). - G. C. Greubel, Nov 30 2024

			Triangle begins as:
        1;
        3,      3;
        0,     24,      12;
      -30,      0,     180,      60;
        0,   -720,       0,    1440,      360;
     2520,      0,  -12600,       0,    12600,   2520;
        0, 120960,       0, -201600,        0, 120960,   20160;
  -771120,      0, 3810240,       0, -3175200,      0, 1270080, 181440;

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

Cf. A001715, A005990, A028296.

m:= 13;
R:=PowerSeriesRing(Integers(), m+1);
EulerE:= func< n | (2^(n+1)/(n+1))*( Evaluate(BernoulliPolynomial(n+1), 1/2) - 2^(n+1)*Evaluate(BernoulliPolynomial(n+1), 1/4) ) >;
f:= func< n,x | (Factorial(n+2)/2)*( 3*x^n - 2*(&+[ Binomial(n,j)*(EulerE(j)/2^j)*(x - 1/2)^(n-j): j in [0..n]]) ) >;
A166553:= func< n,k | Coefficient(R!( f(n,x) ), k) >;
[A166553(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Nov 30 2024

(* first program *)
p[t_]= Exp[x*t](3*Exp[t] - 1)/(Exp[t] + 1);
With[{m=12}, Table[(n!*(n+2)!/2)*CoefficientList[SeriesCoefficient[ Series[p[t], {t,0,m+1}], n], x], {n,0,m}]]//Flatten
(* Second program *)
f[n_, x_]:= (n+2)!*(3*EulerE[n, x+1] - EulerE[n, x])/4;
A166553[n_, k_]:= Coefficient[Series[f[n, x], {x,0,n}], x, k];
Table[A166553[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 30 2024 *)

def f(n,x): return (factorial(n+2)/2)*( 3*x^n - 2*sum( binomial(n,j)*euler_number(j)*(x-1/2)^(n-j)/2^j for j in range(n+1)) )
def A166553(n,k): return ( f(n,x) ).series(x,n+1).list()[k]
print(flatten([[A166553(n,k) for k in range(n+1)] for n in range(14)])) # G. C. Greubel, Nov 30 2024

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (n!*(n+2)!/2) * [t^n]( exp(x*t)*(3*exp(t) - 1)/(exp(t) + 1) ).
From G. C. Greubel, Nov 30 2024: (Start)
T(n, k) = [x^k]( (n+2)!*(3*EulerE(n, x+1) - EulerE(n, x))/4 ).
T(n, k) = [x^k]( (1/2)*(n+2)!*( 3*x^n - 2*Sum_{j=0..n} binomial(n,j)*(EulerE(j)/2^j)*(x - 1/2)^(n-j) ) ).
T(n, n) = 3*A001715(n+2) = (n+2)!/2.
T(n, n-1) = 3*A005990(n+1). (End)

I rewrote the definition. - N. J. A. Sloane, Dec 14 2010

New name by G. C. Greubel, Nov 30 2024

A292930 Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of k-sequences of balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence.

Original entry on oeis.org

1, 2, 8, 3, 24, 60, 4, 48, 240, 480, 5, 80, 600, 2400, 4200, 6, 120, 1200, 7200, 25200, 40320, 7, 168, 2100, 16800, 88200, 282240, 423360, 8, 224, 3360, 33600, 235200, 1128960, 3386880, 4838400, 9, 288, 5040, 60480, 529200, 3386880, 15240960, 43545600, 59875200, 10, 360, 7200, 100800, 1058400, 8467200, 50803200, 217728000, 598752000, 798336000

Offset: 1

Jeremy Dover, Sep 26 2017

Note that the three matching balls are necessarily the same color.

			n=1 => AAA -> T(1,3)=1;
n=2 => AAA,BBB -> T(2,3)=2;
       AAAB,AABA,ABAA,BAAA,BBBA,BBAB,BABB,ABBB -> T(2,4)=8.
Triangle begins:
  1;
  2, 8;
  3, 24, 60;
  4, 48, 240, 480;
  5, 80, 600, 2400, 4200;
  ...

Columns of table: T(n,3) = A000027(n), T(n,4) = A033996(n).

Other sequences in table: T(n,n+2) = A005990(n+1).

T(n, k) = binomial(k,3)*n!/(n+2-k)!;
tabl(nn) = for (n=1, nn, for (k=3, n+2, print1(T(n,k), ", ")); print()); \\ Michel Marcus, Sep 29 2017

T(n, k) = binomial(k,3)*n!/(n+2-k)!.

cp's OEIS Frontend

A062193 Fourth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).

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A324224 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

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A062194 Fifth column sequence of triangle A062139 (generalized a=2 Laguerre).

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GAP

Magma

Mathematica

PARI

Sage

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A138770 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} such that there are exactly k entries between the entries 1 and 2 (n>=2, 0<=k<=n-2).

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Maple

Mathematica

Formula

A167568 A triangle related to the GF(z) formulas of the rows of the ED2 array A167560.

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A317527 Number of edges in the n-alternating group graph.

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Magma

Mathematica

Formula

A062195 Sixth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).

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Magma

Mathematica

PARI

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A193474 Table read by rows: The coefficients of the polynomials P(n, x) = Sum{k=0..n} Sum{j=0..k} (-1)^j * 2^(-k) * binomial(k, j) * (k-2*j)^n * x^(n-k).

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A193474 Table read by rows: The coefficients of the polynomials P(n, x) = Sum{k=0..n} Sum{j=0..k} (-1)^j * 2^(-k) * binomial(k, j) * (k-2j)^n x^(n-k).

A166553 Triangle read by rows: T(n, k) = [x^k]( (n+2)!(3EulerE(n, x+1) - EulerE(n, x))/4 ).