A060187 -id:A060187 - cp's OEIS frontend

A145905 Square array read by antidiagonals: Hilbert transform of triangle A060187.

1, 1, 1, 1, 3, 1, 1, 9, 5, 1, 1, 27, 25, 7, 1, 1, 81, 125, 49, 9, 1, 1, 243, 625, 343, 81, 11, 1, 1, 729, 3125, 2401, 729, 121, 13, 1, 1, 2187, 15625, 16807, 6561, 1331, 169, 15, 1, 1, 6561, 78125, 117649, 59049, 14641, 2197, 225, 17, 1, 1, 19683, 390625, 823543

Offset: 0

Peter Bala, Oct 27 2008

Definition of the Hilbert transform of a triangular array:
For many square arrays in the database the entries in a row are polynomial in the column index, of degree d say and hence the row generating function has the form P(x)/(1-x)^(d+1), where P is some polynomial function. Often the array whose rows are formed from the coefficients of these P polynomials is of independent interest. This suggests the following definition.
Let [L(n,k)]n,k>=0 be a lower triangular array and let R(n,x) := sum {k = 0 .. n} L(n,k)*x^k, denote the n-th row generating polynomial of L. Then we define the Hilbert transform of L, denoted Hilb(L), to be the square array whose n-th row, n >= 0, has the generating function R(n,x)/(1-x)^(n+1).
In this particular case, L is the array A060187, the array of Eulerian numbers of type B, whose row polynomials are the h-polynomials for permutohedra of type B. The Hilbert transform is an infinite Vandermonde matrix V(1,3,5,...).
We illustrate the Hilbert transform with a few examples:
(1) The Delannoy number array A008288 is the Hilbert transform of Pascal's triangle A007318 (view as the array of coefficients of h-polynomials of n-dimensional cross polytopes).
(2) The transpose of the array of nexus numbers A047969 is the Hilbert transform of the triangle of Eulerian numbers A008292 (best viewed in this context as the coefficients of h-polynomials of n-dimensional permutohedra of type A).
(3) The sequence of Eulerian polynomials begins [1, x, x + x^2, x + 4*x^2 + x^3, ...]. The coefficients of these polynomials are recorded in triangle A123125, whose Hilbert transform is A004248 read as square array.
(4) A108625, the array of crystal ball sequences for the A_n lattices, is the Hilbert transform of A008459 (viewed as the triangle of coefficients of h-polynomials of n-dimensional associahedra of type B).
(5) A142992, the array of crystal ball sequences for the C_n lattices, is the Hilbert transform of A086645, the array of h-vectors for type C root polytopes.
(6) A108553, the array of crystal ball sequences for the D_n lattices, is the Hilbert transform of A108558, the array of h-vectors for type D root polytopes.
(7) A086764, read as a square array, is the Hilbert transform of the rencontres numbers A008290.
(8) A143409 is the Hilbert transform of triangle A073107.

			Triangle A060187 (with an offset of 0) begins
1;
1, 1;
1, 6, 1;
so the entries in the first three rows of the Hilbert transform of
A060187 come from the expansions:
Row 0: 1/(1-x) = 1 + x + x^2 + x^3 + ...;
Row 1: (1+x)/(1-x)^2 = 1 + 3*x + 5*x^2 + 7*x^3 + ...;
Row 2: (1+6*x+x^2)/(1-x)^3 = 1 + 9*x + 25*x^2 + 49*x^3 + ...;
The array begins
n\k|..0....1.....2.....3......4
================================
0..|..1....1.....1.....1......1
1..|..1....3.....5.....7......9
2..|..1....9....25....49.....81
3..|..1...27...125...343....729
4..|..1...81...625..2401...6561
5..|..1..243..3125.16807..59049
...

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.
S. Parker, The Combinatorics of Functional Composition and Inversion, Ph.D. Dissertation, Brandeis Univ. (1993) [From _Tom Copeland_, Nov 09 2008]

Cf. A008292, A039755, A052750 (first superdiagonal), A060187, A114172, A145901.

T:=(n,k) -> (2*k + 1)^n: seq(seq(T(n-k,k),k = 0..n),n = 0..10);

T(n,k) = (2*k + 1)^n, (see equation 4.10 in [Franssens]). This array is the infinite Vandermonde matrix V(1,3,5,7, ....) having a LDU factorization equal to A039755 * diag(2^n*n!) * transpose(A007318).

A142175 Triangle read by rows: T(n,k) = (1/4)(A007318(n,k) - 6A008292(n+1,k+1) + 9*A060187(n+1,k+1)).

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 36, 36, 1, 1, 133, 420, 133, 1, 1, 449, 3334, 3334, 449, 1, 1, 1446, 21939, 49364, 21939, 1446, 1, 1, 4534, 130044, 560957, 560957, 130044, 4534, 1, 1, 13991, 724222, 5459561, 10284514, 5459561, 724222, 13991, 1, 1, 42747, 3880014, 48160170, 154214412, 154214412, 48160170, 3880014, 42747, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 16 2008

Row n gives the coefficients in the expansion of (1/4)*(1 + x)^n + (9/4)*2^n*(1 - x)^(1 + n)*Phi(x, -n, 1/2) - (3/2)*(1 - x)^(n + 2)*Phi(x, -1 - n, 1), where Phi is the Lerch transcendant.

			Triangle begins:
     1;
     1,    1;
     1,    8,      1;
     1,   36,     36,      1;
     1,  133,    420,    133,      1;
     1,  449,   3334,   3334,    449,      1;
     1, 1446,  21939,  49364,  21939,   1446,    1;
     1, 4534, 130044, 560957, 560957, 130044, 4534, 1;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Wikipedia, Lerch zeta function
Wikipedia, Polylogarithm

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A168287, A168288, A168289, A168290, A168291, A168292, A168293.

A060187:= func< n,k | (&+[(-1)^(k-j)*Binomial(n, k-j)*(2*j-1)^(n-1): j in [1..k]]) >;
A142175:= func< n,k | (Binomial(n,k) - 6*EulerianNumber(n+1,k) + 9*A060187(n+1,k+1))/4 >;
[A142175(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 30 2024

p[x_, n_] = 1/4*(1 + x)^n + 9/4*2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2] - 3/2*(1 - x)^(2 + n)*PolyLog[-1 - n, x]/x;
Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 0, 10}]// Flatten

A008292(n, k) := sum((-1)^j*(k - j)^n*binomial(n + 1, j), j, 0, k)$
A060187(n, k) := sum((-1)^(k - j)*binomial(n, k - j)*(2*j - 1)^(n - 1), j, 1, k)$
T(n, k) := (binomial(n, k) - 6*A008292(n + 1, k + 1) + 9*A060187(n + 1, k + 1))/4$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 20 2018 */

# from sage.all import * # (use for Python)
from sage.combinat.combinat import eulerian_number
def A060187(n,k): return sum(pow(-1, k-j)*binomial(n, k-j)*pow(2*j-1, n-1) for j in range(1,k+1))
def A142175(n,k): return (binomial(n,k) - 6*eulerian_number(n+1,k) +9*A060187(n+1,k+1))//4
print(flatten([[A142175(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Dec 30 2024

E.g.f.: (exp((1 + x)*y) - 6*(1 - x)^2*exp(y*(1 - x))/(1 - x*exp(y*(1 - x)))^2 + 9*(1 - x)*exp((1 - x)*y)/(1 - x*exp(2*(1 - x)*y)))/4. - Franck Maminirina Ramaharo, Oct 20 2018

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 19 2018

A060188 A column and diagonal of A060187.

Original entry on oeis.org

1, 6, 23, 76, 237, 722, 2179, 6552, 19673, 59038, 177135, 531428, 1594309, 4782954, 14348891, 43046704, 129140145, 387420470, 1162261447, 3486784380, 10460353181, 31381059586, 94143178803, 282429536456, 847288609417

Offset: 2

N. J. A. Sloane, Mar 20 2001

Sums of rows of the numerators and of the denominators of the redundant Stern-Brocot structure A152975/A152976: a(n+2) = Sum_{k=2^n..(2^(n+1) -1)} A152975(k) = Sum_{k=2^n..(2^(n+1) -1)} A152976(k). - Reinhard Zumkeller, Dec 22 2008

Vincenzo Librandi, Table of n, a(n) for n = 2..2000
P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 19 (1920), 305-340; Coll. Papers II, pp. 267-302.
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A048473, A060187 (first differences).

[3^(n-1)-n: n in [2..30]]; // Vincenzo Librandi, Sep 05 2011

a[0]:=1:for n from 1 to 24 do a[n]:=(4*a[n-1]-3*a[n-2]+2) od: seq(a[n], n=0..24); # Zerinvary Lajos, Jun 08 2007

Table[3^(n-1) -n, {n,2,30}] (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
LinearRecurrence[{5,-7,3},{1,6,23},30] (* Harvey P. Dale, Jul 03 2024 *)

[3^(n-1) -n for n in (2..32)] # G. C. Greubel, Jan 07 2022

a(n) = 3^(n-1) - n = A061980(n-1, 2). - Henry Bottomley, May 24 2001
From Paul Barry, Jun 24 2003: (Start)
With offset 0, this is 3^(n+1) - n - 2.
Partial sums of A048473. (End)
From Colin Barker, Dec 19 2012: (Start)
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).
G.f.: x^2*(1 + x)/((1-x)^2*(1-3*x)). (End)
E.g.f.: (exp(3*x) - 3*x*exp(x) - 1)/3. - Wolfdieter Lang, Apr 17 2017

More terms from Vladeta Jovovic, Mar 20 2001

A176490 Triangle T(n,k) = A008292(n+1,k+1) + A060187(n+1,k+1)- 1 read along rows 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 33, 33, 1, 1, 101, 295, 101, 1, 1, 293, 1983, 1983, 293, 1, 1, 841, 11733, 25963, 11733, 841, 1, 1, 2425, 64949, 275341, 275341, 64949, 2425, 1, 1, 7053, 346219, 2573521, 4831203, 2573521, 346219, 7053, 1, 1, 20685, 1804179, 22163163

Offset: 0

Roger L. Bagula, Apr 19 2010

Row sums are: 1, 2, 11, 68, 499, 4554, 51113, 685432, 10684791, 189423350, 3755807989,....

Conjecture on the row sums s(n): 859*(n+1)*s(n) +(-2577*n^2-15955*n+33324)*s(n-1) +(1718*n^3+39275*n^2-102106*n-16383)*s(n-2) +(-25038*n^3+35127*n^2+252701*n-453082)*s(n-3) +(n-3)*(57834*n^2-211893*n+212386)*s(n-4) -2*(17257*n-29530)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 16 2015

			1;
1, 1;
1, 9, 1;
1, 33, 33, 1;
1, 101, 295, 101, 1;
1, 293, 1983, 1983, 293, 1;
1, 841, 11733, 25963, 11733, 841, 1;
1, 2425, 64949, 275341, 275341, 64949, 2425, 1;
1, 7053, 346219, 2573521, 4831203, 2573521, 346219, 7053, 1;
1, 20685, 1804179, 22163163, 70723647, 70723647, 22163163, 1804179, 20685, 1;
1, 61073, 9268777, 180504391, 916661395, 1542816715, 916661395, 180504391, 9268777, 61073, 1;

Cf. A007318, A008292, A060187, A176487.

A176490 := proc(n,k)
    A008292(n+1,k+1)+A060187(n+1,k+1)-1 ;
end proc: # R. J. Mathar, Jun 16 2015

(*A060187*)
p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
<< DiscreteMath`Combinatorica`;
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Eulerian[1 + n, m];
t[n_, m_, 2] := f[n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]

A177043 Central MacMahon numbers: a(n)=A060187(2*n+1, n+1).

Original entry on oeis.org

1, 6, 230, 23548, 4675014, 1527092468, 743288515164, 504541774904760, 455522635895576646, 527896878148304296900, 763820398700983273655796, 1349622683586635111555174216, 2859794140516672651686471055900, 7157996663278223282076538528360968

Offset: 0

Roger L. Bagula, May 01 2010

nonn

Robert Israel, Table of n, a(n) for n = 0..201

Cf. A000108, A060187, A154420.

a:= n-> add((-1)^(n-i) *binomial(2*n+1, n-i) *(2*i+1)^(2*n), i=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 05 2011
# With the generating function of the generalized Eulerian polynomials:
gf := proc(n, k) local f; f := (x,t) -> x*exp(t*x/k)/(1-x*exp(t*x));
series(f(x,t), t, n+2); ((1-x)/x)^(n+1)*k^n*n!*coeff(%, t, n):
collect(simplify(%),x) end: seq(coeff(gf(2*n,2),x,n),n=0..13); # Peter Luschny, May 02 2013

(*A060187*)
p[x_,n_]=(1-x)^(n+1)*Sum[(2*k+1)^n*x^k,{k,0,Infinity}];
f[n_,m_]:=CoefficientList[FullSimplify[ExpandAll[p[x,n]]],x][[m+1]];
a=Table[f[2*n,n],{n,0,20}]

a(n) ~ sqrt(3) * 2^(4*n+1) * n^(2*n) / exp(2*n). - Vaclav Kotesovec, Sep 30 2014

A225356 Triangle T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -22, 1, 1, -75, -75, 1, 1, -236, 1446, -236, 1, 1, -721, 9822, 9822, -721, 1, 1, -2178, 58479, -201244, 58479, -2178, 1, 1, -6551, 325061, -2160227, -2160227, 325061, -6551, 1, 1, -19672, 1736668, -19971304, 49441990, -19971304, 1736668, -19672, 1

Offset: 0

Roger L. Bagula, May 07 2013

			The triangle begins:
  1;
  1,      1;
  1,    -22,       1;
  1,    -75,     -75,         1;
  1,   -236,    1446,      -236,        1;
  1,   -721,    9822,      9822,     -721,         1;
  1,  -2178,   58479,   -201244,    58479,     -2178,       1;
  1,  -6551,  325061,  -2160227, -2160227,    325061,   -6551,      1;
  1, -19672, 1736668, -19971304, 49441990, -19971304, 1736668, -19672, 1;

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

Cf A007318, A060187, A159041.

(* First program *)
q[x_, n_]= (1-x)^(n+1)*Sum[(2*m+1)^n*x^m, {m, 0, Infinity}];
t[n_, m_]:= t[n, m]= Table[CoefficientList[q[x, k], x], {k,0,15}][[n+1, m+1]];
p[x_, n_]:= p[x, n]= Sum[x^i*If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i <= Floor[n/2], (-1)^i*t[n, i], (-1)^(n-i+1)*t[n, i]]], {i,0,n}]/(1-x);
Flatten[Table[CoefficientList[p[x, n], x], {n,10}]]
(* Second Program *)
A060187[n_, k_]:= Sum[(-1)^(k-i)*Binomial[n, k-i]*(2*i-1)^(n-1), {i,k}];
T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], T[n, k-1] +(-1)^k*A060187[n+2, k+1], T[n, n-k] ]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2022 *)

def A060187(n,k): return sum( (-1)^(k-j)*(2*j-1)^(n-1)*binomial(n, k-j) for j in (1..k) )
@CachedFunction
def A225356(n,k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return A225356(n,k-1) + (-1)^k*A060187(n+2,k+1)
    else: return A225356(n,n-k)
flatten([[A225356(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2022

T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1.

Edited by N. J. A. Sloane, May 11 2013

Edited by G. C. Greubel, Mar 18 2022

A060189 A column and diagonal of A060187 (k=3).

Original entry on oeis.org

1, 23, 230, 1682, 10543, 60657, 331612, 1756340, 9116141, 46702427, 237231970, 1198382694, 6031771195, 30287995733, 151856096504, 760614930344, 3807336276505, 19050241098975, 95294209168414, 476607030432890

Offset: 3

N. J. A. Sloane, Mar 20 2001

G. C. Greubel, Table of n, a(n) for n = 3..1000
P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 19 (1920), 305-340; Coll. Papers II, pp. 267-302.
Index entries for linear recurrences with constant coefficients, signature (14,-75,196,-263,174,-45).

Cf. A060187.

[5^(n-1) -n*3^(n-1) +n*(n-1)/2: n in [3..40]]; // G. C. Greubel, Jul 31 2024

Table[5^(n-1) -n*3^(n-1) +n*(n-1)/2, {n,3,40}] (* G. C. Greubel, Jul 31 2024 *)

[5^(n-1) -n*3^(n-1) +n*(n-1)//2 for n in range(3,41)] # G. C. Greubel, Jul 31 2024

a(n) = 5^(n-1) - n*3^(n-1) + n*(n-1)/2. - Ralf Stephan, May 08 2004
G.f.: x^3*(1 + 9*x - 17*x^2 - 9*x^3) / ((1-x)^3*(1-3*x)^2*(1-5*x)). - Colin Barker, Dec 19 2012
From Wolfdieter Lang, Apr 17 2017: (Start)
a(n) = A060187(n, 3) , n >= 3 (and 0 for n = 0,1,2).
a(n) = A060187(n, n-2), n >= 3 (and 0 for n = 0,1,2).
E.g.f.: (2*exp(5*x) - 10*x*exp(3*x) + 5*x^2*exp(x) - 2)/10. (End)

More terms from Vladeta Jovovic, Mar 20 2001

A060190 A column and diagonal of A060187 (k=4).

Original entry on oeis.org

1, 76, 1682, 23548, 259723, 2485288, 21707972, 178300904, 1403080725, 10708911188, 79944249686, 587172549764, 4261002128223, 30644790782352, 218917362275080, 1556000598766224, 11017646288488233, 77790282457881756

Offset: 4

N. J. A. Sloane, Mar 20 2001

G. C. Greubel, Table of n, a(n) for n = 4..1000
P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 19 (1921), 305-340; Coll. Papers II, pp. 267-302.
Index entries for linear recurrences with constant coefficients, signature (30,-385,2776,-12418,35908,-67818,82552,-62109,26190,-4725).

Cf. A060187.

[(&+[(-1)^k*Binomial(n,k)*(7-2*k)^(n-1): k in [0..3]]): n in [4..40]]; // G. C. Greubel, Aug 01 2024

r := proc(n, k) option remember;
if n = 0 then if k = 0 then 1 else 0 fi else
(2*(n-k)+1)*r(n-1, k-1) + (2*k+1)*r(n-1, k) fi end:
A060189 := n -> r(n-1, 3): seq(A060189(n), n = 4..21); # Peter Luschny, May 06 2013

r[n_, k_] := r[n, k] = If[n == 0, If[k == 0, 1, 0], (2*(n-k)+1)*r[n-1, k-1] + (2*k+1)*r[n-1, k]]; A060189[n_] := r[n-1, 3]; Table[A060189[n], {n, 4, 21}] (* Jean-François Alcover, Dec 03 2013, translated from Peter Luschny's program *)
A060190[n_]:= Sum[(-1)^k*Binomial[n,k]*(7-2*k)^(n-1), {k,0,3}];
Table[A060190[n], {n,4,40}] (* G. C. Greubel, Aug 01 2024 *)

[sum((-1)^k*binomial(n,k)*(7-2*k)^(n-1) for k in range(4)) for n in range(4,40)] # G. C. Greubel, Aug 01 2024

From Wolfdieter Lang, Apr 17 2017: (Start)
a(n) = A060187(n, 4), n >= 4, and 0 for n < 4,
a(n) = A060187(n, n-3), n >= 4, and 0 for n < 4.
O.g.f.: x^4*(1 + 46*x - 213*x^2 - 428*x^3 + 2295*x^4 - 1794*x^5 - 675*x^6) / Product_{j=0..3} (1 - (1+2*j)*x)^(4-j).
E.g.f.: (exp(7*x) - 7*x*exp(5*x) + (21*x^2/2)*exp(3*x) - (7*x^3/3!)*exp(x) - 1)/7. (End)
a(n) = Sum_{k=0..3} (-1)^k*binomial(n,k)*(7-2*k)^(n-1). - G. C. Greubel, Aug 01 2024

More terms from Vladeta Jovovic, Mar 20 2001

A143213 Triangle T(n,m) read by rows: Gray code of A060187(n, k) (decimal representation), 1 <= k <= n, n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 28, 28, 1, 1, 106, 149, 106, 1, 1, 155, 987, 987, 155, 1, 1, 955, 440, 514, 440, 955, 1, 1, 194, 137, 974, 974, 137, 194, 1, 1, 340, 754, 60, 293, 60, 754, 340, 1, 1, 181, 238, 166, 377, 377, 166, 238, 181, 1, 1, 977, 283, 540, 411, 142, 411, 540, 283, 977, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 20 2008

			Triangle begins as:
  1;
  1,   1;
  1,   5,   1;
  1,  28,  28,   1;
  1, 106, 149, 106,   1;
  1, 155, 987, 987, 155,   1;
  1, 955, 440, 514, 440, 955,   1;
  1, 194, 137, 974, 974, 137, 194,   1;
  1, 340, 754,  60, 293,  60, 754, 340,   1;
  1, 181, 238, 166, 377, 377, 166, 238, 181,   1;
  1, 977, 283, 540, 411, 142, 411, 540, 283, 977,  1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Eric Weisstein, Mathematica Notebook GrayCode.nb
Eric Weisstein, Gray Code, MathWorld.

Cf. A060187, A143214.

GrayCode[n_, k_]:= FromDigits[BitXor@@@Partition[Prepend[IntegerDigits[n,2,k], 0], 2, 1], 2];
A060187[n_, k_]:= Sum[(-1)^(k-j)*Binomial[n,k-j]*(2*j-1)^(n-1), {j,k}];
A143213[n_, k_]:= GrayCode[A060187[n, k], 10];
Table[A143213[n,k], {n,12}, {k,n}]//Flatten

T(n, n-k) = T(n, k). - G. C. Greubel, Aug 08 2024

Edited by G. C. Greubel, Aug 27 2024

A143506 Irregular triangle read by rows: first row is 1, and n-th row gives the coefficients of x^(n - 1)*R(n,x + 1/x)/(x + 1/x), where R(n,x) is the n-th row polynomial for A060187.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 3, 6, 1, 1, 23, 26, 47, 26, 23, 1, 1, 76, 234, 304, 467, 304, 234, 76, 1, 1, 237, 1687, 2630, 5293, 4787, 5293, 2630, 1687, 237, 1, 1, 722, 10549, 27158, 52730, 78586, 84365, 78586, 52730, 27158, 10549, 722, 1, 1, 2179, 60664, 272797, 563029, 1132234

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 25 2008

Row sums yield A080253.

			Triangle begins:
   1;
   1,   1,    1;
   1,   6,    3,    6,    1;
   1,  23,   26,   47,   26,   23,    1;
   1,  76,  234,  304,  467,  304,  234,   76,    1;
   1, 237, 1687, 2630, 5293, 4787, 5293, 2630, 1687, 237, 1;
    ... reformatted. - _Franck Maminirina Ramaharo_, Oct 25 2018

Wikipedia, Lerch zeta function

Cf. A008292, A060187.

Cf. A143505, A143507.

Table[CoefficientList[FullSimplify[ExpandAll[2^n*(1 - x - 1/x)^(1 + n)*x^n*LerchPhi[x + 1/x, -n, 1/2]]], x], {n, 0, 10}]//Flatten

Row n is generated by the polynomial 2^n*(1 - x - 1/x)^(1 + n)*x^n*Phi(x + 1/x, -n, 1/2), where Phi is the Lerch transcendant.

E.g.f.: (1 - x + x^2)*exp((1 + x + x^2)*t)/((1 + x^2)*exp(2*t*x) - x*exp(2*(1 + x^2)*t)). - Franck Maminirina Ramaharo, Oct 25 2018

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 25 2018

cp's OEIS Frontend

A145905 Square array read by antidiagonals: Hilbert transform of triangle A060187.

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A142175 Triangle read by rows: T(n,k) = (1/4)*(A007318(n,k) - 6*A008292(n+1,k+1) + 9*A060187(n+1,k+1)).

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A060188 A column and diagonal of A060187.

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A176490 Triangle T(n,k) = A008292(n+1,k+1) + A060187(n+1,k+1)- 1 read along rows 0<=k<=n.

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A177043 Central MacMahon numbers: a(n)=A060187(2*n+1, n+1).

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A225356 Triangle T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows.

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A060189 A column and diagonal of A060187 (k=3).

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A142175 Triangle read by rows: T(n,k) = (1/4)(A007318(n,k) - 6A008292(n+1,k+1) + 9*A060187(n+1,k+1)).