A156922 -id:A156922 - cp's OEIS frontend

A000340 a(0)=1, a(n) = 3*a(n-1) + n + 1.

1, 5, 18, 58, 179, 543, 1636, 4916, 14757, 44281, 132854, 398574, 1195735, 3587219, 10761672, 32285032, 96855113, 290565357, 871696090, 2615088290, 7845264891, 23535794695, 70607384108, 211822152348, 635466457069

Offset: 0

N. J. A. Sloane, Simon Plouffe

From Johannes W. Meijer, Feb 20 2009: (Start)
Second right hand column (n-m=1) of the A156920 triangle.
The generating function of this sequence enabled the analysis of the polynomials A156921 and A156925.
(End)
Partial sums of A003462, and thus the second partial sums of A000244 (3^n). Also column k=2 of A106516. - John Keith, Jan 04 2022

			G.f. = 1 + 5*x + 18*x^2 + 58*x^3 + 179*x^4 + 543*x^5 + 1636*x^6 + ...

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See pp. 9, 18.
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See p. 7.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 389
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. Mentions this sequence. See also Proc. Amer. Math. Soc. 148 (2020), 461-469.
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

From Johannes W. Meijer, Feb 20 2009: (Start)
Cf. A156921, A156925, A156927, A156933. Other columns A156922, A156923, A156924.
Equals A156920 second right hand column.
Equals A142963 second right hand column divided by 2^n.
Equals A156919 second right hand column divided by 2.
(End)
Cf. A014915.
Equals column k=1 of A008971 (shifted). - Jeremy Dover, Jul 11 2021
Cf. A000340, A003462 (first differences), A106516.

[(3^(n+2)-2*n-5)/4: n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-3*a[n-2]+1 od: seq(a[n],n=0..50); # Miklos Kristof, Mar 09 2005
A000340:=-1/(3*z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

a[ n_] := MatrixPower[ {{1, 0, 0}, {1, 1, 0}, {1, 1, 3}}, n + 1][[3, 1]]; (* Michael Somos, May 28 2014 *)
RecurrenceTable[{a[0]==1,a[n]==3a[n-1]+n+1},a,{n,30}] (* or *) LinearRecurrence[{5,-7,3},{1,5,18},30] (* Harvey P. Dale, Jan 31 2017 *)

G.f.: 1/((1-3*x)*(1-x)^2).
a(n) = (3^(n+2) - 2*n - 5)/4.
a(n) = Sum_{k=0..n+1} (n-k+1)*3^k = Sum_{k=0..n+1} k*3^(n-k+1). - Paul Barry, Jul 30 2004
a(n) = Sum_{k=0..n} binomial(n+2, k+2)*2^k. - Paul Barry, Jul 30 2004
a(-1)=0, a(0)=1, a(n) = 4*a(n-1) - 3*a(n-2) + 1. - Miklos Kristof, Mar 09 2005
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3). - Johannes W. Meijer, Feb 20 2009
a(-2 - n) = 3^-n * A014915(n). - Michael Somos, May 28 2014
E.g.f.: exp(x)*(9*exp(2*x) - 2*x - 5)/4. - Stefano Spezia, Nov 09 2024

A156919 Table of coefficients of polynomials related to the Dirichlet eta function.

Original entry on oeis.org

1, 2, 1, 4, 10, 1, 8, 60, 36, 1, 16, 296, 516, 116, 1, 32, 1328, 5168, 3508, 358, 1, 64, 5664, 42960, 64240, 21120, 1086, 1, 128, 23488, 320064, 900560, 660880, 118632, 3272, 1, 256, 95872, 2225728

Offset: 0

Johannes W. Meijer, Feb 20 2009, Jun 24 2009

Essentially the same as A185411. Row reverse of A185410. - Peter Bala, Jul 24 2012
The SF(z; n) formulas, see below, were discovered while studying certain properties of the Dirichlet eta function.
From Peter Bala, Apr 03 2011: (Start)
Let D be the differential operator 2*x*d/dx. The row polynomials of this table come from repeated application of the operator D to the function g(x) = 1/sqrt(1 - x). For example,
  D(g) = x*g^3
  D^2(g) = x*(2 + x)*g^5
  D^3(g) = x*(4 + 10*x + x^2)*g^7
  D^4(g) = x*(8 + 60*x + 36*x^2 + x^3)*g^9.
Thus this triangle is analogous to the triangle of Eulerian numbers A008292, whose row polynomials come from the  repeated application of the operator x*d/dx to the function 1/(1 - x). (End)

			The first few rows of the triangle are:
  [1]
  [2, 1]
  [4, 10, 1]
  [8, 60, 36, 1]
  [16, 296, 516, 116, 1]
The first few P(z;n) are:
  P(z; n=0) = 1
  P(z; n=1) = 2 + z
  P(z; n=2) = 4 + 10*z + z^2
  P(z; n=3) = 8 + 60*z + 36*z^2 + z^3
The first few SF(z;n) are:
  SF(z; n=0) = (1/2)*(1)/(1-z)^(3/2);
  SF(z; n=1) = (1/4)*(2+z)/(1-z)^(5/2);
  SF(z; n=2) = (1/8)*(4+10*z+z^2)/(1-z)^(7/2);
  SF(z; n=3) = (1/16)*(8+60*z+36*z^2+z^3)/(1-z)^(9/2);
In the Savage-Viswanathan paper, the coefficients appear as
  1;
  1,    2;
  1,   10,     4;
  1,   36,    60,     8;
  1,  116,   516,   296,    16;
  1,  358,  3508,  5168,  1328,   32;
  1, 1086, 21120, 64240, 42960, 5664, 64;
  ...

D. H. Lehmer, Interesting Series Involving the Central Binomial Coefficient, Am. Math. Monthly 92 (1985) 449-457, Polynomial V in eq (17). [R. J. Mathar, Feb 24 2009]
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, arXiv: 1204.4963v3 [math.CO], 2012.
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11
S.-M. Ma and T. Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, arXiv preprint arXiv:1409.6525 [math.CO], 2014.
S.-M. Ma and Y.-N. Yeh, Stirling permutations, cycle structures of permutations and perfect matchings, arXiv preprint arXiv:1503.06601 [math.CO], 2015.
Carla D. Savage and Gopal Viswanathan, The 1/k-Eulerian polynomials, Elec. J. of Comb., Vol. 19, Issue 1, #P9 (2012).
Eric Weisstein's World of Mathematics, Dirichlet Eta Function

A142963 and this sequence can be mapped onto the A156920 triangle.
FP1 sequences A000340, A156922, A156923, A156924.
FP2 sequences A050488, A142965, A142966, A142968.
Appears in A162005, A000182, A162006 and A162007.
Cf. A186695, A187075.
Cf. A185410 (row reverse), A185411.

A156919 := proc(n,m) if n=m then 1; elif m=0 then 2^n ; elif m<0 or m>n then 0; else 2*(m+1)*procname(n-1,m)+(2*n-2*m+1)*procname(n-1,m-1) ; end if; end proc: seq(seq(A156919(n,m), m=0..n), n=0..7); # R. J. Mathar, Feb 03 2011

g[0] = 1/Sqrt[1-x]; g[n_] := g[n] = 2x*D[g[n-1], x]; p[n_] := g[n] / g[0]^(2n+1) // Cancel; row[n_] := CoefficientList[p[n], x] // Rest; Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Aug 09 2012, after Peter Bala *)
Flatten[Table[Rest[CoefficientList[Nest[2 x D[#, x] &, (1 - x)^(-1/2), k] (1 - x)^(k + 1/2), x]], {k, 10}]] (* Jan Mangaldan, Mar 15 2013 *)

SF(z; n) = Sum_{m >= 1} m^(n-1)*4^(-m)*z^(m-1)*Gamma(2*m+1)/(Gamma(m)^2) = P(z;n) / (2^(n+1)*(1-z)^((2*n+3)/2)) for n >= 0. The polynomials P(z;n) = Sum_{k = 0..n} a(k)*z^k generate the a(n) sequence.
If we write the sequence as a triangle the following relation holds: T(n,m) = (2*m+2)*T(n-1,m) + (2*n-2*m+1)*T(n-1,m-1) with T(n,m=0) = 2^n and T(n,n) = 1, n >= 0 and 0 <= m <= n.
G.f.: 1/(1-xy-2x/(1-3xy/(1-4x/(1-5xy/(1-6x/(1-7xy/(1-8x/(1-... (continued fraction). - Paul Barry, Jan 26 2011
From Peter Bala, Apr 03 2011 (Start)
E.g.f.: exp(z*(x + 2)) * (1 - x)/(exp(2*x*z) - x*exp(2*z))^(3/2) = Sum_{n >= 0} P(x,n)*z^n/n! = 1 + (2 + x)*z + (4 + 10*x + x^2)*z^2/2! + (8 + 60*x + 36*x^2 + x^3)*z^3/3! + ... .
Explicit formula for the row polynomials:
P(x,n-1) = Sum_{k = 1..n} 2^(n-2*k)*binomial(2k,k)*k!*Stirling2(n,k)*x^(k-1)*(1 - x)^(n-k).
The polynomials x*(1 + x)^n * P(x/(x + 1),n) are the row polynomials of A187075.
The polynomials x^(n+1) * P((x + 1)/x,n) are the row polynomials of A186695.
Row sums are A001147(n+1). (End)
Sum_{k = 0..n} (-1)^k*T(n,k) = (-1)^binomial(n,2)*A012259(n+1). - Johannes W. Meijer, Sep 27 2011

Minor edits from Johannes W. Meijer, Sep 27 2011

A156920 Triangle of the normalized A142963 and A156919 sequences.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 15, 18, 1, 1, 37, 129, 58, 1, 1, 83, 646, 877, 179, 1, 1, 177, 2685, 8030, 5280, 543, 1, 1, 367, 10002, 56285, 82610, 29658, 1636, 1, 1, 749, 34777, 335162, 919615, 756218, 159742, 4916, 1

Offset: 0

Johannes W. Meijer, Feb 20 2009

The originator sequences are A142963 and A156919.
The Flower Triangle seems to be an appropriate name for the triangular array of this sequence. The zero patterns of the Flower Polynomials of the first, see A156921, the second, see A156925, the third, see A156927, and the fourth kind, see A156933, look like flowers.
The first Maple program generates the Flower Triangle sequence.
The second program generates the Right Hand Columns sequences and the third one generates the Left Hand Column sequences. For an explanation of these two algorithms see A142963.

			The first few rows of the triangle are:
  [1]
  [1, 1]
  [1, 5, 1]
  [1, 15, 18, 1]
  [1, 37, 129, 58, 1]
  [1, 83, 646, 877, 179, 1]

Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO]. - From N. J. A. Sloane, Aug 21 2012

Originator sequences A142963, A156919.
Related sequences A156921, A156925, A156927, A156933.
Left hand column sequences A050488, A142965, A142966, A142968.
Right hand column sequences A000340, A156922, A156923, A156924.
Row sums A014307(n+1).

A156920 := proc(n,m): if n=m then 1; elif m=0 then 1 ; elif m<0 or m>n then 0; else (m+1)*procname(n-1, m)+(2*n-2*m+1)*procname(n-1, m-1) ; end if; end proc: seq(seq(A156920(n, m), m=0..n), n=0..8);
RHCnr:=5; RHCmax:=10; RHCend:=RHCnr+RHCmax: for k from RHCnr to RHCend do for n from 0 to k do S2[k,n]:=sum((-1)^(n+i)*binomial(n,i)*i^k/n!,i=0..n) end do: G(k,x):= sum(S2[k,p]*((2*p)!/p!) *x^p/(1-4*x)^(p+1),p=0..k)/(((-1)^(k+1)*2*x)/(-1+4*x)^(k+1)): fx:=simplify(G(k,x)): nmax:=degree(fx); RHC[k-RHCnr+1]:= coeff(fx,x,k-RHCnr)/2^(k-RHCnr) end do: a:=n-> RHC[n]: seq(a(n), n=1..RHCend-RHCnr);
LHCnr:=5; LHCmax:=10: LHCend:=LHCnr+LHCmax: for k from LHCnr to LHCend do for n from 0 to k do S2[k,n]:=sum((-1)^(n+i)*binomial(n,i)*i^k/n!,i=0..n) end do: G(k,x):= sum(S2[k,p]*((2*p)!/p!)*x^p/(1-4*x)^(p+1),p=0..k)/ (((-1)^(k+1)*2*x)/(-1+4*x)^(k+1)): fx:=simplify(G(k,x)): nmax:=degree(fx); for n from 0 to nmax do d[n]:= coeff(fx,x,n)/2^n end do: LHC[n]:=d[LHCnr-1] end do: a:=n-> LHC[n]: seq(a(n), n=LHCnr..LHCend-1);

T[, 0] = 1; T[n, n_] = 1; T[n_, m_] := T[n, m] = (m + 1)*T[n - 1, m] + (2*n - 2*m + 1)*T[n - 1, m - 1];
Table[T[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2017 *)

T(n,m) = (m+1)*T(n-1,m) + (2*n-2*m+1)*T(n-1,m-1) with T(n,m=0) = 1 and T(n,n) = 1, n>=0 and 0 <= m <= n.
From Peter Bala, Jul 22 2012: (Start)
T(n,k) = 1/(2^(n-k))*A156919(n,k).
E.g.f.: 1 + t*x + (t+t^2)*x^2/2! + (t+5*t^2+t^3)*x^3/3! + ... = sqrt(E(x,2*t)), where E(x,t) = (1-t)*exp(x*t)/(exp(x*t)-t*exp(x)) = 1 + t*x + (t+t^2)*x^2/2! + (t+4*t^2+t^3)*x^3/3! + ... is the e.g.f. for the Eulerian numbers A008292.
The row polynomials R(n,x) satisfy 1/sqrt(1-2*x)*(x*d/dx)^n(1/sqrt(1-2*x)) = R(n,x)/(1-2*x)^(n+1). (End)

Minor edits by Johannes W. Meijer, Sep 28 2011

A142963 Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)x^p(1-4x)^(k-1-p)(2*p+2)!/(p+1)!.

Original entry on oeis.org

1, 1, 2, 1, 10, 4, 1, 30, 72, 8, 1, 74, 516, 464, 16, 1, 166, 2584, 7016, 2864, 32, 1, 354, 10740, 64240, 84480, 17376, 64, 1, 734, 40008, 450280, 1321760, 949056, 104704, 128, 1, 1498, 139108, 2681296, 14713840, 24198976, 10223488, 629248, 256, 1, 3030, 462264, 14341992

Offset: 1

Wolfdieter Lang, Sep 15 2008

Previous name: Table of coefficients of row polynomials of certain o.g.f.s.
The o.g.f.s G(k, x) for the k-family of sequences S(k, n):= Sum_{p=0..n} p^k*binomial(2*p, p)*binomial(2*(n-p), n-p), k=0,1,... (convolution of two sequences involving the central binomial coefficients) are 1/(1-4*x) for k=0 and 2*x*P(k, x)/(1-4*x)^(k+1) for k=1,2,..., with the row polynomials P(k, x) = Sum_{m=0..k-1} a(n,m)*x^m).
The author was led to compute the sums S(k, n) by a question asked by M. Greiter, Jun 27 2008.
In order to keep the index k>=1 of Sigma(k, n) also for the polynomials P(k, x), their degree is then k-1.

			Triangle starts:
[1]
[1,   2]
[1,  10,     4]
[1,  30,    72,      8]
[1,  74,   516,    464,      16]
[1, 166,  2584,   7016,    2864,     32]
[1, 354, 10740,  64240,   84480,  17376,     64]
[1, 734, 40008, 450280, 1321760, 949056, 104704, 128]
...
P(3,x) = 1+10*x+4*x^2.
G(3,x) = 2*x*(1+10*x+4*x^2)/(1-4*x)^4.

Vincenzo Librandi, Table of n, a(n) for n = 1..210
Wolfdieter Lang, First 10 rows and more.
L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207v5 [math.CO], 2005-2006.

Left hand column sequences 2*A142964, 4*A142965, 8*A142966, 16*A142968.
Row sums A142967.
From Johannes W. Meijer, Feb 20 2009: (Start)
A156919 and this sequence can be mapped onto A156920.
Cf. A156921, A156925, A156927, A156933.
Right hand column sequences 2^n*A000340, 2^n*A156922, 2^n*A156923, 2^n*A156924. (End)
Cf. A142961, A142962.

A142963 := proc(n,m): if n=m+1 then 2^(n-1); elif m=0 then 1 ; elif m<0 or m>n-1 then 0; else (m+1)*procname(n-1, m)+(4*n-4*m-2)*procname(n-1, m-1); end if; end proc: seq(seq(A142963(n,m), m=0..n-1), n=1..9); # Johannes W. Meijer, Sep 28 2011
# Alternatively (assumes offset 0):
p := (n,x) -> (1/2)*add(Stirling2(n+1,k+1)*x^k*(1-4*x)^(n-k)*(2*k+2)!/(k+1)!, k=0..n): for n from 0 to 7 do [n], PolynomialTools:-CoefficientList(p(n,x), x) od;
# Peter Luschny, Jun 18 2017

t[, 0] = 1; t[n, m_] /; m == n-1 := 2^m; t[n_, m_] := (m+1)*t[n-1, m] + (4*n-4*m-2)*t[n-1, m-1]; Table[t[n, m], {n, 1, 10}, {m, 0, n-1}] // Flatten (* Jean-François Alcover, Jun 21 2013, after Johannes W. Meijer *)

G(k, x) = Sum_{p=0..k} S2(k, p)*((2*p)!/p!)*x^p/(1-4*x)^(p+1), k >= 0 (here k >= 1), with the Stirling2 triangle S2(k, p):=A048993(k, p). (Proof from the product of the o.g.f.s of the two convoluted sequences and the normal ordering (x^d_x)^k = Sum_{p=0..k} S2(k, p)*x^p*d_x^p, with the derivative operator d_x.)
a(k,m) = [x^m]P(k, x) = [x^m] ((1-4*x)^(k+1))*G(k,x)/(2*x), k>=1, m=0,1,...,k-1.
For the triangle coefficients the following relation holds: T(n,m) = (m+1)*T(n-1,m) + (4*n-4*m-2)*T(n-1,m-1) with T(n,m=0) = 1 and T(n,m=n-1) = 2^(n-1), n >= 1 and 0 <= m <= n-1. - Johannes W. Meijer, Feb 20 2009
From Peter Bala, Jan 18 2018: (Start)
(x*d/dx)^n (1/(sqrt(1 - 4*x)) = 2*x*P(n,x)/sqrt(1 - 4*x)^(n+1/2) for n >= 1.
x*P(n,x)/(1 - 4*x)^(n+1/2) = (1/2)*Sum_{k >= 1} binomial(2*k,k)* k^n*x^k for n >= 1.
P(n+1,x) = ((4*n - 2)*x + 1)*P(n,x) - x*(4*x - 1)*d/dx(P(n,x)).
Hence the polynomial P(n,x) has all real zeros by Liu et al., Theorem 1.1, Corollary 1.2. (End)

Minor edits by Johannes W. Meijer, Sep 28 2011
A more precise name by Peter Luschny, Jun 18 2017
Name reformulated with offset corrected, edited by Wolfdieter Lang, Aug 23 2019

A156921 FP1 polynomials related to the generating functions of the right hand columns of the A156920 triangle.

Original entry on oeis.org

1, 1, 1, 1, -6, 1, 7, -79, 119, 126, -270, 1, 28, -515, 1654, 8689, -65864, 142371, -82242, -99090, 113400, 1, 86, -2255, 5784, 300930, -3904584, 20663714, -41517272, -80232259, 657717054

Offset: 0

Johannes W. Meijer, Feb 20 2009

The FP1 polynomials appear in the numerators of the GF1 o.g.f.s. of the right hand columns of A156920. The FP1 can be calculated with the formula for the RHC sequence, see A156920, and the formula for the general structure of the generating function GF1, see below.
An appropriate name for the FP1 polynomials seems to be the flower polynomials of the first kind because the zero patterns of these polynomials look like flowers. The zero patterns of the FP2, see A156925, and the FP1 resemble each other closely.
A Maple program that generates for a right hand column with a certain RHCnr its GF1 and FP1 can be found below. RHCnr stands for right hand column number and starts from 1.

			The first few rows of the "triangle" of the coefficients of the FP1 polynomials.
In the columns the coefficients of the powers of z^m, m=0,1,2,... , appear.
  [1]
  [1]
  [1, 1, -6]
  [1, 7, -79, 119, 126, -270]
  [1, 28, -515, 1654, 8689, -65864, 142371, -82242, -99090, 113400]
Matrix of the coefficients of the FP1 polynomials. The coefficients in the columns of this matrix are the powers of z^m, m=0,1,2,.. .
  [1, 0 ,0, 0, 0, 0, 0, 0, 0, 0]
  [1, 0 ,0, 0, 0, 0, 0, 0, 0, 0]
  [1, 1, -6, 0 ,0, 0, 0, 0, 0, 0]
  [1, 7, -79, 119, 126, -270, 0, 0, 0, 0]
  [1, 28, -515, 1654, 8689, -65864, 142371, -82242, -99090, 113400]
The first few FP1 polynomials are:
  FP1(z; RHCnr=1) = 1
  FP1(z; RHCnr=2) = 1
  FP1(z; RHCnr =3) = 1+z-6*z^2
Some GF1(z;RHCnr) are:
  GF1(z;RHCnr= 3) = (1+z-6*z^2)/((1-5*z)*(1-3*z)^2*(1-z)^3)
  GF1(z;RHCnr= 4) = (1+7*z-79*z^2+119*z^3+126*z^4-270*z^5)/((1-7*z)*(1-5*z)^2*(1-3*z)^3*(1-z)^4)

Cf. A156920, A156925, A156927, A156933.
For the first few GF1's see A000340, A156922, A156923, A156924.
The number of FP1 terms follow the triangular numbers A000217, with quite surprisingly one exception here a(0)=1.
Abs(Row sums (n)) = A098695(n).
For the polynomials in the denominators of the GF1(z;RHCnr) see A157702.

RHCnr:=4: if RHCnr=1 then RHCmax :=1; else RHCmax:=(RHCnr-1)*(RHCnr)/2 end if: RHCend:=RHCnr+RHCmax: for k from RHCnr to RHCend do for n from 0 to k do S2[k,n]:=sum((-1)^(n+i)*binomial(n,i)*i^k/n!,i=0..n) end do: G(k,x):= sum(S2[k,p]*((2*p)!/p!) *x^p/(1-4*x)^(p+1),p=0..k)/(((-1)^(k+1)*2*x)/(-1+4*x)^(k+1)): fx:=simplify(G(k,x)): nmax:=degree(fx); RHC[k-RHCnr+1]:= coeff(fx,x,k-RHCnr)/2^(k-RHCnr) end do: a:=n-> RHC[n]: seq(a(n), n=1..RHCend-RHCnr+1); for nx from 0 to RHCmax do num:=sort(sum(A[t]*z^t, t=0..RHCmax)); nom:=Product((1-(2*u-1)*z)^(RHCnr-u+1),u=1..RHCnr): RHCa:= series(num/nom,z,nx+1); y:=coeff(RHCa,z,nx)-A[nx]; x:=RHC[nx+1]; A[nx]:=x-y; end do: FP1[RHCnr]:=sort(num,z, ascending); GenFun[RHCnr] :=FP1[RHCnr]/product((1-(2*m-1)*z)^(RHCnr-m+1),m=1..RHCnr);

G.f.: GF1(z;RHCnr) := FP1(z;RHCnr)/product((1-(2*m-1)*z)^(RHCnr+1-m),m=1..RHCnr)

Row sums (n) = (-1)^(1+(n+1)*(n+2)/2)*A098695(n).

A156923 Fourth right hand column (n-m=3) of the A156920 triangle.

Original entry on oeis.org

1, 37, 646, 8030, 82610, 756218, 6411720, 51586344, 400011435, 3020658295, 22373863774, 163379472214, 1180488191108, 8462445970580, 60305767988960, 427848087263712, 3025286818472661

Offset: 0

Johannes W. Meijer, Feb 20 2009

Other columns A000340, A156922, A156924.
Equals A156920 fourth right hand column.
Equals A156919 fourth right hand column divided by 8.
Equals A142963 fourth right hand column divided by 2^n

a(n) = 30*a(n-1)-385*a(n-2)+2776*a(n-3)-12418*a(n-4)+35908*a(n-5)-67818*a(n-6)+82552*a(n-7)-62109*a(n-8)+26190*a(n-9)-4725*a(n-10)
a(n) = (-8*n^3+972*n^2*3^n-84*n^2+7776*n*3^n-11250*n*5^n-286*n+15309*3^n-50625*5^n+36015*7^n-315)/384
G.f.: GF1(z;RHCnr=4) = (1+7*z-79*z^2+119*z^3+126*z^4-270*z^5)/((1-7*z)*(1-5*z)^2*(1-3*z)^3*(1-z)^4)

A156924 Fifth right hand column (n-m=4) of the A156920 triangle.

Original entry on oeis.org

1, 83, 2685, 56285, 919615, 12813843, 160206627, 1854550395, 20291056470, 212826091180, 2161547322134, 21414479565774, 208076662576370, 1991164206775450, 18825064380813450

Offset: 0

Johannes W. Meijer, Feb 20 2009

Other columns A000340, A156922, A156923.
Equals A156920 fifth right hand column.
Equals A156919 fifth right hand column divided by 16.
Equals A142963 fifth right hand column divided by 2^n

a(n)=55*a(n-1)-1365*a(n-2)+20251*a(n-3)-200557*a(n-4)+1402203*a(n-5)-7137473*a(n-6)+26886431*a(n-7)-75433971*a(n-8)+157376597*a(n-9)-241846607*a(n-10)+268663713*a(n-11)-208880991*a(n-12)+107416665*a(n-13)-32730075*a(n-14)+4465125*a(n-15)
a(n)= (16*n^4-7776*n^3*3^n+256*n^3-104976*n^2*3^n+225000*n^2*5^n+ 1496*n^2- 464616*n*3^n+ 2250000*n*5^n-2016840*n*7^n+3776*n-673596*3^n+5568750*5^n-11092620*7^n+6200145*9^n+3465)/6144
G.f.: GF1(z;RHCnr=5) = (1+28*z-515*z^2+1654*z^3+8689*z^4-65864*z^5+142371*z^6-82242*z^7-99090*z^8+113400*z^9)/((1-9*z)*(1-7*z)^2*(1-5*z)^3*(1-3*z)^4*(1-z)^5)

A102365 Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 18, 15, 1, 0, 1, 58, 129, 37, 1, 0, 1, 179, 877, 646, 83, 1, 0, 1, 543, 5280, 8030, 2685, 177, 1, 0, 1, 1636, 29658, 82610, 56285, 10002, 367, 1, 0, 1, 4916, 159742, 756218, 919615, 335162, 34777, 749, 1, 0

Offset: 0

Philippe Deléham, Feb 22 2005

Generalized Eulerian numbers A008292.

Reversal of A211399. - Philippe Deléham, Feb 12 2013

			Triangle begins:
  1;
  1,  0;
  1,  1,   0;
  1,  5,   1,  0;
  1, 18,  15,  1, 0;
  1, 58, 129, 37, 1, 0; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Diagonals: A000007, A000012, A050488, A142965, A142966.

Columns: A000012, A000340, A156922, A156923, A156924.

T[0, 0] := 1;  T[n_, -1] := 0;  T[n_, n_] := 0; T[n_, k_] := T[n, k] = (n - k)*T[n - 1, k - 1] + (2*k + 1)*T[n - 1, k]; Join[{1}, Table[If[k < 0, 0, If[k >= n, 0, T[n, k]]], {n, 1, 5}, {k, 0, n}] // Flatten] (* G. C. Greubel, Jun 30 2017 *)

T(n, k) = (n-k)*T(n-1, k-1) + (2*k+1)*T(n-1, k) with T(0, 0) = 1, T(0, k) = 0 if k > 0, T(n, k) = 0 if k < 0.
Sum_{k>=0} T(n, k)*2^k = A001147(n).
Sum_{k>=0} T(n, k) = A014307(n). - Philippe Deléham, Mar 19 2005

A211399 Triangle T(n,k), 0 <= k <= n, given by (0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...) DELTA (1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 15, 18, 1, 0, 1, 37, 129, 58, 1, 0, 1, 83, 646, 877, 179, 1, 0, 1, 177, 2685, 8030, 5280, 543, 1, 0, 1, 367, 10002, 56285, 82610, 29658, 1636, 1, 0, 1, 749, 34777, 335162

Offset: 0

Philippe Deléham, Feb 08 2013

Contains A156920 as submatrix.

Row-reversal of A102365. - Philippe Deléham, Feb 12 2013

			Triangle begins :
1
0, 1
0, 1, 1
0, 1, 5, 1
0, 1, 15, 18, 1
0, 1, 37, 129, 58, 1
0, 1, 83, 646, 877, 179, 1

Left hand column sequences: A000007, A000012, A050488, A142965, A142966, A142968.
Right hand column sequences: A000340, A156922, A156923, A156924.
Row sums A014307(n).

T(n,k) = k*T(n-1,k) + (2n-2k+1)*T(n-1,k-1) , T(n,n) = 1, T(n,k) = 0 if k<0 or if k>n.
T(n,k) = A185411(n,k)/(2^(n-k)).
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000012(n), A014307(n), A001147(n) for x = 0, 1, 2 respectively .
G.f.: 1/(1-xy/(1-x/(1-3xy/(1-2x/(1-5xy/(1-3x/(1-7xy/(1- ...(continued fraction).

cp's OEIS Frontend

A000340 a(0)=1, a(n) = 3*a(n-1) + n + 1.

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A156919 Table of coefficients of polynomials related to the Dirichlet eta function.

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A156920 Triangle of the normalized A142963 and A156919 sequences.

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A142963 Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!.

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A156921 FP1 polynomials related to the generating functions of the right hand columns of the A156920 triangle.

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A156923 Fourth right hand column (n-m=3) of the A156920 triangle.

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A156924 Fifth right hand column (n-m=4) of the A156920 triangle.

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A102365 Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.

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A142963 Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)x^p(1-4x)^(k-1-p)(2*p+2)!/(p+1)!.