A060188 -id:A060188 - cp's OEIS frontend

A060187 Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2n - 2k + 1)T(n-1, k-1) + (2k - 1)*T(n-1, k).

1, 1, 1, 1, 6, 1, 1, 23, 23, 1, 1, 76, 230, 76, 1, 1, 237, 1682, 1682, 237, 1, 1, 722, 10543, 23548, 10543, 722, 1, 1, 2179, 60657, 259723, 259723, 60657, 2179, 1, 1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1, 1, 19673, 1756340, 21707972, 69413294, 69413294, 21707972, 1756340, 19673, 1

Offset: 1

N. J. A. Sloane, Mar 20 2001

Rows are expansions of p(x,n) = 2^n*(1 - x)^(1 + n)*LerchPhi(x, -n, 1/2). Row sums are A000165. - Roger L. Bagula, Sep 16 2008
Eulerian numbers of type B. The n-th row of this triangle is the h-vector of the simplicial complex dual to a permutohedron of type B_(n-1). For example, the permutohedron of type B_2 is an octagon whose dual, also an octagon, has f-polynomial f(x) = 1 + 8*x + 8*x^2 and h-polynomial given by (x-1)^2 + 8*(x-1) + 8 = 1 + 6*x + x^2, giving [1,6,1] as row 3 of this table (see Fomin and Reading, p. 21). The corresponding triangle of f-vectors for the type B permutohedra is A145901. The Hilbert transform of the current array is A145905. - Peter Bala, Oct 26 2008
From Peter Bala, Oct 13 2011: (Start)
The row polynomials count the elements of the hyperoctahedral group B_n (the group of signed permutations on n letters) according to the number of type B descents (see Chow and Gessel).
Let P denote Pascal's triangle. Then the first column of the array P*(I-t*P^2)^(-1) (I the identity array) begins [1/(1-t),(1+t)/(1-t)^2,(1+6*t+t^2)/(1-t)^3,...]. The numerator polynomials are the row polynomials of this table. Similarly, in the array (I-t*A062715)^-1, the numerator polynomials in the first column produce the row polynomials of this table (but with an extra factor of t). Cf. A145901. (End)
The Dasse-Hartaut and Hitczenko paper (section 6.1.4) shows this triangle of numbers, when suitably normalized, satisfies the central limit theorem. - Peter Bala, Mar 05 2012
These are the coefficients of the midpoint Eulerian polynomials (see Quade/Collatz and Schoenberg). In terms of the cardinal B-splines b_n(t) these polynomials can be defined as M_n(x) = 2^n*n!*Sum_{k=0..n} b_{n+1}(k+1/2)*x^k. - Peter Luschny, Apr 26 2013
The o.g.f. Godd(n, x) = Sum_{m>=0} Sodd(n, m)*x^m, with Sodd(n, m) = Sum_{j=0..m} (1+2*j)^n is Podd(n, x)/(1 - x)^(n+2) with Podd(n, x) = Sum_{k=0..n} T(n+1, k+1)*x^k. E.g., Godd(2, x) = (1 + 6*x + x^2)/(1 - x)^4; see A000447(n+1) for n >= 0. For the e.g.f.s see A282628. - Wolfdieter Lang, Mar 17 2017
Let h_0(x,y) = x*y/(x+y), and D = x*D_x - y*D_y where D_x is the partial derivative w.r.t. x, etc.  Put h_{n+1}(x,y) = D(h_n)(x,y).  Then h_n(x,y) = x*y/(x+y)^(n+1)*f_{n}(x,y) where f_n(x,y) = Sum_{k=0..n} (-1)^k*T(n+1,k+1)*y^(n-k)*x^k.  If instead of h_0, one similarly uses g_0(x,y) = x*y/(y-x), etc., then one obtains g_n(x,y) = x*y/(y-x)^(n+1)*Sum_{k=0..n} T(n+1,k+1)*y^(n-k)*x^k. (If instead of D one considers D' = x*D_x + y*D_y, then h_0 and g_0 are fixed points of D'.) - Gregory Gerard Wojnar, Oct 28 2018
Counts coloop-free Schubert delta-matroids by cornered rank, see Remark 4.6 of the paper by Eur, Fink, Larson, Spink. - Matt Larson, May 20 2024

			The triangle T(n, k) begins:
n\k 1    2     3      4      5     6    7 8 ...
1:  1
2:  1    1
3:  1    6     1
4:  1   23    23      1
5:  1   76   230     76      1
6:  1  237  1682   1682    237     1
7:  1  722 10543  23548  10543   722    1
8:  1 2179 60657 259723 259723 60657 2179 1
...
row n = 9: 1 6552 331612 2485288 4675014 2485288 331612 6552 1,
row n = 10: 1 19673 1756340 21707972 69413294 69413294 21707972 1756340 19673 1,
row n = 11: 1 59038 9116141 178300904 906923282 1527092468 906923282 178300904 9116141 59038 1, ... reformatted. - _Wolfdieter Lang_, Mar 17 2017

G. Boros and V. H. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, 2004.
T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 11.
W. Quade and L. Collatz, Zur Interpolationstheorie der reellen periodischen Funktionen. Sitzungsbericht der Preuss. Akad. der Wiss., Phys.-Math. Kl, (1938), 383-429.

Muniru A Asiru, Table of n, a(n) for n = 1..1275
Juan Arias de Reyna, Integral Representation for Riemann-Siegel Z(t) function, arXiv:2406.18968 [math.NT], 2024. See p. 11.
Jean-Christophe Aval, Adrien Boussicault, and Philippe Nadeau, Tree-like Tableaux, Electronic Journal of Combinatorics, 20(4), 2013, #P34.
Eli Bagno and David Garber, Type-B analogue of Bell numbers using Rota's Umbral calculus approach, arXiv:2406.16393 [cs.DM], 2024. See p. 5.
Eli Bagno, David Garber, and Mordechai Novick, The Worpitzky identity for the groups of signed and even-signed permutations, arXiv:2004.03681 [math.CO], 2020.
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
Jose Bastidas, The polytope algebra of generalized permutahedra, arXiv:2009.05876 [math.CO], 2020.
Victor Batyrev and Mark Blume, On generalizations of Losev-Manin moduli systems for classical root systems arXiv:0912.2898 [math.AG], 2009-2011, (p. 13). - _Tom Copeland_, Oct 03 2014
Anna Borowiec and Wojciech Mlotkowski, New Eulerian numbers of type D, arXiv:1509.03758 [math.CO], 2015.
Chak-On Chow and I. M. Gessel, On the descent numbers and major indices for the hyperoctahedral group, Adv. Appl. Math. 38, No. 3, 275-301 (2007).
Sandrine Dasse-Hartaut and Pawel Hitczenko, Greek letters in random staircase tableaux, arXiv:1202.3092 [math.CO], 2012.
Andrew Ducharme, Unifying trigonometric and hyperbolic function derivatives via negative integer order polylogarithms, arXiv:2405.19371 [math.GM], 2024. See p. 6.
Christopher Eur, Alex Fink, Matt Larson, and Hunter Spink, Signed permutohedra, delta-matroids, and beyond, arXiv:2209.06752 [math.AG], 2022-2024; Proc. Lond. Math. Soc. 3 (2024). Paper No. e12592, 54pp.
Sergey Fomin and Nathan 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.
Pawel Hitczenko and Svante Janson, Weighted random staircase tableaux, arXiv:1212.5498 [math.CO], 2012.
Svante Janson, Euler-Frobenius numbers and rounding, arXiv:1305.3512 [math.PR], 2013.
Zakhar Kabluchko and Hugo Panzo, A refinement of the Sylvester problem: Probabilities of combinatorial types, arXiv:2501.16166 [math.PR], 2025. See p. 24.
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.
Lily L. Liu and Yi Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.
Peter Luschny, Eulerian polynomials.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - From N. J. A. Sloane, Aug 21 2012
Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013.
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11.
Shi-Mei Ma, Qi Fang, Toufik Mansour, and Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374, 2021
Shi-Mei Ma, Jun Ma, and Yeong-Nan Yeh, On certain combinatorial expansions of descent polynomials and the change of grammars, arXiv:1802.02861 [math.CO], 2018.
Shi-Mei Ma, T. Mansour, and D. Callan, Some combinatorial arrays related to the Lotka-Volterra system, arXiv:1404.0731 [math.CO], 2014.
Shi-Mei Ma and Hai-Na Wang, Enumeration of a dual set of Stirling permutations by their alternating runs, arXiv:1506.08716 [math.CO], 2015.
P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 19 (1921), 305-340; Coll. Papers II, pp. 267-302.
Fumihiko Nakano and Taizo Sadahiro, A generalization of carries process and Eulerian numbers, arXiv:1306.2790 [math.PR], 2013.
G. Rzadkowski, An Analytic Approach to Special Numbers and Polynomials, J. Int. Seq. 18 (2015) 15.8.8.
I. J. Schoenberg, Cardinal interpolation and spline functions IV. The exponential Euler splines. ISNM 20 (1972), 382-404.
Richard P. Stanley and Fabrizio Zanello, Unimodality of partitions with distinct parts inside Ferrers shapes, arXiv:1305.6083 [math.CO], 2013.
Richard P. Stanley and Fabrizio Zanello, Some asymptotic results on q-binomial coefficients, 2014.
Einar Steingrímsson, Permutation statistics of indexed permutations, European J. Combin. 15 (1994), no. 2, 187-205.
G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_2(n,k).

Diagonals give A060188, A060189, A060190. Cf. A008292.

Cf. also A000165 (row sums), A002436 (alt. row sums), A008292, A145901, A145905 (Hilbert transform). A062715.

a:=Flat(List([1..11],n->List([1..n],k->Sum([1..k],i->(-1)^(k-i)*Binomial(n,k-i)*(2*i-1)^(n-1))))); # Muniru A Asiru, Feb 09 2018

[[(&+[(-1)^(k-j)*Binomial(n,k-j)*(2*j-1)^(n-1): j in [1..k]]): k in [1..n]]: n in [1..10]]; // G. C. Greubel, Nov 08 2018

A060187:= (n,k) -> add((-1)^(k-i)*binomial(n,k-i)*(2*i-1)^(n-1), i = 1..k):
for n from 1 to 10 do seq(A060187(n,k),k = 1..n); end do; # Peter Bala, Oct 26 2008
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 10 do lprint([seq(T(n,k,2),k=1..n)]); od; # N. J. A. Sloane, May 08 2013
P := proc(n,x) option remember; if n = 0 then 1 else
  (n*x+(1/2)*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
  expand(%) fi end:
A060187 := (n,k) -> 2^n*coeff(P(n,x),x,k):
seq(print(seq(A060187(n,k), k=0..n)), n=0..10);  # Peter Luschny, Mar 08 2014

p[x_, n_] = 2^n (1 - x)^(1 + n) LerchPhi[x, -n, 1/2]; Table[CoefficientList[p[x, n], x], {n, 0, 10}] // Flatten (* Roger L. Bagula, Sep 16 2008 *)
T[n_, k_] := Sum[(-1)^(k-i)*Binomial[n, k-i]*(2*i-1)^(n-1), {i, 1, k}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 23 2015, after Peter Bala *)

{T(n, k) = if( nMichael Somos, Jan 07 2011 */

from math import isqrt, comb
def A060187(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    b = n-comb(a,2)
    return sum(-comb(a,b-i)*((i<<1)-1)**(a-1) if b-i&1 else comb(a,b-i)*((i<<1)-1)**(a-1) for i in range(1,b+1)) # Chai Wah Wu, Nov 13 2024

@CachedFunction
def A060187(n, k) :
    if n == 0: return 1 if k == 0 else 0
    return (2*(n-k)+1)*A060187(n-1, k-1) + (2*k+1)*A060187(n-1, k)
for n in (0..8): [A060187(n,k) for k in (0..n)] # Peter Luschny, Apr 26 2013

T(s, 2) = 3^(s-1) - s. Sum_{t=1..s} T(s, t) = 2^(s-1)*(s-1)!.
From Peter Bala, Oct 26 2008: (Start)
T(n,k) = Sum_{i = 1..k} (-1)^(k-i)*binomial(n,k-i)*(2*i-1)^(n-1).
E.g.f.: (1 - x)*exp((1 - x)*t)/(1 - x*exp(2*(1 - x)*t)) = 1 + (1 + x)*t + (1 + 6*x + x^2)*t^2/2! + ... .
The row polynomials R(n,x) satisfy R(n,x)/(1 - x)^n = Sum_{i >= 1} (2*i - 1)^(n-1)*x^i. For example, row 3 gives (x + 6*x^2 + x^3)/ (1 - x)^3 = x + 3^2*x^2 + 5 ^2*x^3 + 7^2*x^4 + ... .
The recurrence relation R(n+1,x) = [(2*n+1)*x - 1]*R(n,x) + 2*x*(1 - x)*R'(n,x) shows that the row polynomials R(n,x) have only real zeros (apply Corollary 1.2 of [Liu and Wang]).
Worpitzky-type identity: Sum_{k = 1..n} T(n,k)*binomial(x+k-1,n-1) = (2*x+1)^(n-1).
The nonzero alternating row sums are (-1)^(n-1)*A002436(n). (End)
exp(x)*(d/dx)^n [exp(x)/(1 - exp(2*x))] = R(n+1,exp(2*x))/ (1 - exp(2*x))^(n+1).
Compare with Example 12.3.1. in [Boros and Moll]. - Peter Bala, Nov 07 2008
The n-th row polynomial R(n,x) = Sum_{k = 0..n} A145901(n,k)*x^k*(1 - x)^(n-k) = Sum_{k = 0..n} A145901(n,k)*(x - 1)^(n-k). - Peter Bala, Jul 22 2014
Assuming an offset 0, the n-th row polynomial = (x - 1)^n * log(x) * Integral_{u = 0..inf} (2*floor(u) + 1)^n * x^(-u) du, provided x > 1. - Peter Bala, Feb 06 2015
The finite sums of consecutive odd integer powers is derived from this number triangle: Sum_{k=1..n}(2k-1)^m = Sum_{j=1..m+1}binomial(n+m+1-j,m+1)*T(m+1,j). - Tony Foster III, Feb 09 2018

A152975 Numerators of the redundant Stern-Brocot structure; denominators=A152976.

Original entry on oeis.org

1, 1, 3, 2, 1, 3, 2, 5, 3, 6, 3, 1, 3, 2, 5, 3, 6, 3, 7, 4, 9, 5, 10, 5, 9, 4, 1, 3, 2, 5, 3, 6, 3, 7, 4, 9, 5, 10, 5, 9, 4, 9, 5, 12, 7, 15, 8, 15, 7, 14, 7, 15, 8, 15, 7, 12, 5, 1, 3, 2, 5, 3, 6, 3, 7, 4, 9, 5, 10, 5, 9, 4, 9, 5, 12, 7, 15, 8, 15, 7, 14, 7, 15, 8, 15, 7, 12, 5, 11, 6, 15, 9, 20, 11, 21

Offset: 1

Reinhard Zumkeller, Dec 22 2008

The redundant Stern-Brocot structure is constructed row by row: insert between consecutive terms of the full Stern-Brocot tree their mediant (non-reduced), where the mediant of s/t and u/v = (s+u)/(t+v);
a(2^n-n+2*k) = A007305(2^(n-1)+k+2) for 0<=k<2^(n-1);
a(2^n-n+2*k-1) = A007305(2^(n-1)+k-1+2) + A007305(2^(n-1)+k+2) for 0the graph of this structure describes an interesting ternary representation of the positive rational numbers;
A060188(k+2) = Sum(a(i): 2^k <= i < 2^(k+1)).

			[0/1] . . . . . . . . . . . . . . . . . . . . . . . . . . . [1/0]
.............................. 1/1
............. 1/2 ............ 3/3 ............ 2/1
..... 1/3 ... 3/6 .... 2/3 ... 5/5 ... 3/2 .... 6/3 ... 3/1
. 1/4 3/9 2/5 5/10 3/5 6/9 3/4 7/7 4/3 9/6 5/3 10/5 5/2 9/3 4/1.

Milad Niqui, Formalising Exact Arithmetic, Ph.D. thesis, Radboud Universiteit Nijmegen, IPA Dissertation Series 2004-10, 2.6, p.65f .

Milad Niqui, Formalising Exact Arithmetic
Index entries for sequences related to Stern's sequences

A152976 Denominators of the redundant Stern-Brocot structure; numerators=A152975.

Original entry on oeis.org

1, 2, 3, 1, 3, 6, 3, 5, 2, 3, 1, 4, 9, 5, 10, 5, 9, 4, 7, 3, 6, 3, 5, 2, 3, 1, 5, 12, 7, 15, 8, 15, 7, 14, 7, 15, 8, 15, 7, 12, 5, 9, 4, 9, 5, 10, 5, 9, 4, 7, 3, 6, 3, 5, 2, 3, 1, 6, 15, 9, 20, 11, 21, 10, 21, 11, 24, 13, 25, 12, 21, 9, 18, 9, 21, 12, 25, 13, 24, 11, 21, 10, 21, 11, 20, 9, 15, 6

Offset: 1

Reinhard Zumkeller, Dec 22 2008

a(2^n-n+2*k) = A047679(2^(n-1)+k) for 0<=k<2^(n-1);

a(2^n-n+2*k-1) = A047679(2^(n-1)+k-1) + A047679(2^(n-1)+k) for 0

A060188(k+2) = Sum(a(i): 2^k <= i < 2^(k+1)).

Milad Niqui, Formalising Exact Arithmetic, Ph.D. thesis, Radboud Universiteit Nijmegen, IPA Dissertation Series 2004-10, 2.6, p.65f .

Milad Niqui, Formalising Exact Arithmetic

A085354 a(n) = 34^n - (n+4)2^(n-1).

Original entry on oeis.org

1, 7, 36, 164, 704, 2928, 11968, 48448, 195072, 783104, 3138560, 12567552, 50298880, 201256960, 805158912, 3220914176, 12884246528, 51538231296, 206155546624, 824627691520, 3298522300416, 13194113318912, 52776503607296

Offset: 0

Paul Barry, Jun 24 2003

Binomial transform of A060188.

The depth i nodes of a perfect binary tree are numbered 2^i through 2^(i+1) - 1, so that the root has number 1, depth 1 nodes have numbers 2 and 3, depth 2 nodes have numbers 4, 5, 6 and 7 and so on. We sum all the numbers in the path connecting a leaf node to the root. For a height n tree, a(n) is the sum of these sums for all leaves nodes. So for instance a height 1 tree has paths 1, 2 and 1, 3 connecting the root to the leaves, and (1+2) + (1+3) = a(1) = 7. This interpretation suggests a recursive formula for computing a(n) by completing the paths covered in a(n-1) and adding the leaves. - Jean M. Morales, Oct 24 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-20,16).

Cf. A005187, A060188.

[3*4^n-(n+4)*2^(n-1): n in [0..30]]; // Vincenzo Librandi, Sep 05 2011

Table[3 * 4^n - (n + 4) * 2^(n - 1), {n, 0, 19}] (* Alonso del Arte, Oct 23 2013 *)
LinearRecurrence[{8,-20,16},{1,7,36},30] (* Harvey P. Dale, Apr 08 2019 *)

a(n) = Sum_{m = 2^n..2^(n+1)} A005187(m). a(n) = 2^n*(2^(n+1)-1) + Sum_{k = 0..(n-1)} a(k). - Philippe Deléham, Feb 19 2004
G.f.: (1-x)/((1-4*x)*(1-2*x)^2). - Bruno Berselli, Sep 05 2011
a(n) = 2*a(n-1) + 3*2^(2n-1) - 2^(n-1), a(0) = 1. - Jean M. Morales, Oct 24 2013

A233295 Riordan array ((1+x)/(1-x)^3, 2*x/(1-x)).

Original entry on oeis.org

1, 4, 2, 9, 10, 4, 16, 28, 24, 8, 25, 60, 80, 56, 16, 36, 110, 200, 216, 128, 32, 49, 182, 420, 616, 560, 288, 64, 64, 280, 784, 1456, 1792, 1408, 640, 128, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256, 100, 570, 2160, 5712, 10752, 14400, 13440, 8320, 3072, 512

Offset: 0

Philippe Deléham, Dec 07 2013

Subtriangle of the triangle in A208532.
Row sums are A060188(n+2).
Diagonal sums are A000295(n+2)=A125128(n+1)=A130103(n+2).

			Triangle begins :
1
4, 2
9, 10, 4
16, 28, 24, 8
25, 60, 80, 56, 16
36, 110, 200, 216, 128, 32
49, 182, 420, 616, 560, 288, 64
64, 280, 784, 1456, 1792, 1408, 640, 128
81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256
100, 570, 2160, 5712, 10752, 14400, 13440, 8320, 3072, 512

Cf. Columns: A000290, A006331, A112742.

Cf. Diagonal: A000079.

G.f. for the column k: 2^k*(1+x)/(1-x)^(k+3).
T(n,k) = 2^k*(binomial(n,k)+3*binomial(n,k+1)+2*binomial(n,k+2)), 0<=k<=n.
T(n,0) = 2*T(n-1,0)-T(n-2,0)+2, T(n,k)=2*T(n-1,k)+2*T(n-1,k-1)-2*T(n-2,k-1)-T(n-2,k) for k>=1, T(0,0)=1, T(1,0)=4, T(1,1)=2, T(n,k)=0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k) = A060188(n+2).
Sum_{k=0..n} T(n,k)*(-1)^k = n+1.
T(n,k) = 2*sum_{j=1..n-k+1} T(n-j,k-1).
T(n,k) = 2^k*A125165(n,k).
T(n,n) = 2^n=A000079(n).
T(n,0) = (n+1)^2=A000290(n+1).
exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(16 + 28*x + 24*x^2/2! + 8*x^3/3!) = 16 + 60*x + 200*x^2/2! + 616*x^3/3! + 1792*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), 2*x/(1 - x) ). Cf. A125165. - Peter Bala, Dec 21 2014

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A060187 Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).

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A061980 Square array A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1, read by antidiagonals.

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A156918 Triangle formed by coefficients of the expansion of p(x,n) = (1+x-x^2)^(n+1)*Sum_{j >= 0} (2*j+1)^n*(-x + x^2)^j.

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A152975 Numerators of the redundant Stern-Brocot structure; denominators=A152976.

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A152976 Denominators of the redundant Stern-Brocot structure; numerators=A152975.

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A085354 a(n) = 3*4^n - (n+4)*2^(n-1).

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Magma

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A233295 Riordan array ((1+x)/(1-x)^3, 2*x/(1-x)).

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A060187 Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2n - 2k + 1)T(n-1, k-1) + (2k - 1)*T(n-1, k).

A156918 Triangle formed by coefficients of the expansion of p(x,n) = (1+x-x^2)^(n+1)Sum_{j >= 0} (2j+1)^n*(-x + x^2)^j.

A085354 a(n) = 34^n - (n+4)2^(n-1).