A156927 -id:A156927 - cp's OEIS frontend

A157704 G.f.s of the z^p coefficients of the polynomials in the GF3 denominators of A156927.

1, 1, 5, 32, 186, 132, 10, 56, 2814, 17834, 27324, 11364, 1078, 10, 48, 17988, 494720, 3324209, 7526484, 6382271, 2004296, 203799, 4580, 5, 16, 72210, 7108338, 146595355, 1025458635, 2957655028, 3828236468

Offset: 0

Johannes W. Meijer, Mar 07 2009

The formula for the PDGF3(z;n) polynomials in the GF3 denominators of A156927 can be found below.
The general structure of the GFKT3(z;p) that generate the z^p coefficients of the PDGF3(z; n) polynomials can also be found below. The KT3(z;p) polynomials in the numerators of the GFKT3(z; p) have a nice symmetrical structure.
The sequence of the number of terms of the first few KT3(z;p) polynomials is 1, 2, 4, 7, 10, 13, 14, 17, 20, 23, 26, 29, 32, 34, 36, 39, 42. The differences of this sequence and that of the number of terms of the KT4(z;p), see A157705, follow a simple pattern.
A Maple algorithm that generates relevant GFKT3(z;p) information can be found below.

			Some PDGF3 (z;n) are:
  PDGF3(z;n=3) = (1-z)*(1-2*z)^4*(1-3*z)^7*(1-4*z)^10
  PDGF3(z;n=4) = (1-z)*(1-2*z)^4*(1-3*z)^7*(1-4*z)^10*(1-5*z)^13
The first few GFKT3's are:
  GFKT3(z;p=0) = 1/(1-z)
  GFKT3(z;p=1) = -(5*z+1)/(1-z)^4
  GFKT3(z;p=2) = z*(32+186*z+132*z^2+10*z^3)/(1-z)^7
Some KT3(z,p) polynomials are:
  KT3(z;p=2) = 32+186*z+132*z^2+10*z^3
  KT3(z;p=3) = 56+2814*z+17834*z^2+27324*z^3+11364*z^4+1078*z^5+10*z^6

Originator sequence A156927.
See A002414 for the z^1 coefficients and A157707 for the z^2 coefficients divided by 2.
Row sums equal A064350 and those of A157705.
Cf. A157702, A157703, A157705.

p:=2; fn:=sum((-1)^(n1+1)*binomial(3*p+1,n1) *a(n-n1),n1=1..3*p+1): fk:=rsolve(a(n) = fn,a(k)): for n2 from 0 to 3*p+1 do fz(n2):=product((1-(k+1)*z)^(1+3*k), k=0..n2): a(n2):= coeff(fz(n2),z,p); end do: b:=n-> a(n): seq(b(n), n=0..3*p+1); a(n)=fn; a(k)=sort(simplify(fk)); GFKT3(p):=sum((fk)*z^k, k=0..infinity); q3:=ldegree((numer(GFKT3(p)))): KT3(p):=sort((-1)^(p)*simplify((GFKT3(p)*(1-z)^(3*p+1))/z^q3),z, ascending);

PDGF3(z;n) = Product_{k=0..n} (1-(k+1)*z)^(1+3*k) with n = 1, 2, 3, ...
GFKT3(z;p) = (-1)^(p)*(z^q3)*KT3(z, p)/(1-z)^(3*p+1) with p = 0, 1, 2, ...
The recurrence relation for the z^p coefficients a(n) is a(n) = Sum_{k=1..3*p+1} (-1)^(k+1)*binomial(3*p + 1, k)*a(n-k) with p = 0, 1, 2, ... .

A156932 Row sums of the FP3 polynomials of A156927.

Original entry on oeis.org

1, 2, -80, -107520, 10218700800, 128549621071872000, -362522272186301939712000000, -359926496710320918207041372160000000000, 186528815537358278677575816954481496555520000000000000

Offset: 0

Johannes W. Meijer, Feb 20 2009

The row sums of the FP3 have simple prime factors.

Row sums of A156927.

A157707 The z^2 coefficients of the polynomials in the GF3 denominators of A156927 divided by 2.

Original entry on oeis.org

16, 205, 1165, 4415, 13055, 32606, 72030, 144930, 270930, 477235, 800371, 1288105, 2001545, 3017420, 4430540, 6356436, 8934180, 12329385, 16737385, 22386595, 29542051, 38509130, 49637450, 63324950

Offset: 1

Johannes W. Meijer, Mar 07 2009

See A157704 for background information.

Cf. A156927, A157704.

nmax:=24; for n from 0 to nmax do fz(n):=product((1-(k+1)*z)^(1+3*k),k=0..n); c(n):= coeff(fz(n),z,2)/2; end do: a:=n-> c(n): seq(a(n), n=1..nmax);

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)
a(n) = 1/4*n^6+7/4*n^5+37/8*n^4+34/6*n^3+25/8*n^2+7/12*n
G.f.: (16 + 93*z + 66*z^2 + 5*z^3)/(1-z)^7

A002414 Octagonal pyramidal numbers: a(n) = n(n+1)(2*n-1)/2.

Original entry on oeis.org

1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, 1386, 1794, 2275, 2835, 3480, 4216, 5049, 5985, 7030, 8190, 9471, 10879, 12420, 14100, 15925, 17901, 20034, 22330, 24795, 27435, 30256, 33264, 36465, 39865, 43470, 47286, 51319, 55575, 60060, 64780

Offset: 1

N. J. A. Sloane

Number of ways of covering a 2n X 2n lattice with 2n^2 dominoes with exactly 2 horizontal dominoes. - Yong Kong (ykong@curagen.com), May 06 2000
Equals binomial transform of [0, 1, 7, 6, 0, 0, 0, ...]. - Gary W. Adamson, Jun 14 2008, corrected Oct 25 2012
Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF3 denominators of A156927. See A157704 for background information. - Johannes W. Meijer, Mar 07 2009
This sequence is related to A000326 by a(n) = n*A000326(n) - Sum_{i=0..n-1} A000326(i) and this is the case d=3 in the identity n*(n*(d*n-d+2)/2)-Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Apr 21 2010
2*a(n) gives the principal diagonal of the convolution array A213819. - Clark Kimberling, Jul 04 2012
Partial sums of the figurate octagonal numbers A000567. For each sequence with a linear recurrence with constant coefficients, the values reduced modulo some constant m generate a periodic sequence. For this sequence, these Pisano periods have length 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, ... for m >= 1. - Ant King, Oct 26 2012
Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016
On a square grid of side length n+1, the number of embedded rectangles (where each side is greater than 1). For example, in a 2 X 2 square there is one rectangle, in a 3 X 3 square there are nine rectangles, etc. - Peter Woodward, Nov 26 2017
a(n) is the sum of the numbers in the n X n square array A204154(n). - Ali Sada, Jun 21 2019
Sum of all multiples of n and n+1 that are <= n^2. - Wesley Ivan Hurt, May 25 2023

			a(2) = 9 since there are 9 ways to cover a 4 X 4 lattice with 8 dominoes, 2 of which is horizontal and the other 6 are vertical. - Yong Kong (ykong@curagen.com), May 06 2000
G.f. = x + 9*x^2 + 30*x^3 + 70*x^4 + 135*x^5 + 231*x^6 + 364*x^7 + 540*x^8 + 765*x^9 + ...

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.
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).

T. D. Noe, Table of n, a(n) for n = 1..1000
B. Berselli, A description of the transform in Comments lines: website Matem@ticamente (in Italian).
M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
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
Index entries for sequences related to dominoes
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A004003, A160378.
Cf. A093563 (( 6, 1) Pascal, column m=3). A000567 (differences).
Cf. A156927, A157704. - Johannes W. Meijer, Mar 07 2009
Cf. A000326.
Cf. similar sequences listed in A237616.
Cf. A260234 (largest prime factor of a(n+1)).

[n*(n+1)*(2*n-1)/2: n in [1..50]]; // Vincenzo Librandi, May 24 2016

A002414 := n-> 1/2*n*(n+1)*(2*n-1): seq(A002414(n), n=1..100);

LinearRecurrence[{4,-6,4,-1},{1,9,30,70},40] (* Harvey P. Dale, Apr 12 2013 *)

{a(n) = (2*n - 1) * n * (n + 1) / 2} \\ Michael Somos, Mar 17 2011

a(n) = odd numbers * triangular numbers = (2*n-1)* binomial(n+1,2). - Xavier Acloque, Oct 27 2003
G.f.: x*(1+5*x)/(1-x)^4. [Conjectured by Simon Plouffe in his 1992 dissertation.]
a(n) = A000578(n) + A000217(n-1). - Kieren MacMillan, Mar 19 2007
a(-n) = -A160378(n). - Michael Somos, Mar 17 2011
From Ant King, Oct 26 2012: (Start)
a(n) = a(n-1) + n*(3*n-2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = n*A000326(n) - A002411(n-1), see Berselli's comment.
a(n) = (n+1)*(2*A000567(n)+n)/6.
a(n) = A000292(n) + 5*A000292(n-1) = binomial(n+2,3)+5*binomial(n+1,3).
a(n) = A002413(n) + A000292(n-1).
a(n) = A000217(n) + 6*A000292(n-1).
Sum_{n>=1} 1/a(n) = 2*(4*log(2)-1)/3 = 1.1817258148265...
(End)
a(n) = Sum_{i=0..n-1} (n-i)*(6*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: x*(2 + 7*x + 2*x^2)*exp(x)/2. - Ilya Gutkovskiy, May 23 2016
a(n) = A080851(6,n-1). - R. J. Mathar, Jul 28 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(Pi + 1 - 4*log(2))/3. - Amiram Eldar, Jul 02 2020

More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000

Incorrect formula deleted by Ant King, Oct 04 2012

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

Original entry on oeis.org

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

A156925 FP2 polynomials related to the generating functions of the left hand columns of the A156920 triangle.

Original entry on oeis.org

1, 1, 1, 1, 8, -11, -6, 1, 38, -108, -242, 839, -444, -180, 1, 144, -425, -7382, 48451, -96764, -2559, 257002, -312444, 88344, 30240, 1, 487, 720, -130472, 1277794, -4193514, -6504496

Offset: 0

Johannes W. Meijer, Feb 20 2009

The FP2 polynomials appear in the numerators of the GF2 o.g.f.s. of the left hand columns of A156920. The FP2 can be calculated with the formula of the LHC sequence, see A156920, and the formula for the general structure of the generating function GF2, see below.
An appropriate name for the FP2 polynomials seems to be the flower polynomials of the second kind because the zero patterns of these polynomials look like flowers. The zero patterns of the FP2 and the FP1, see A156921, resemble each other closely.
A Maple program that generates for a left hand column with a certain LHCnr its GF2 and FP2 can be found below. LHCnr stands for left hand column number and starts from 1.

			The first few rows of the "triangle" of the coefficients of the FP2 polynomials.
In the columns the coefficients of the powers of z^m, m=0,1,2,..., appear.
  [1]
  [1, 1]
  [1, 8, -11, -6]
  [1, 38, -108, -242, 839, -444, -180]
  [1, 144, -425, -7382, 48451, -96764, -2559, 257002, -312444, 88344, 30240]
Matrix of the coefficients of the FP2 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]
  [1, 1, 0, 0, 0, 0, 0]
  [1, 8 , -11, -6, 0, 0, 0]
  [1, 38, -108, -242, 839, -444, -180]
The first few FP2 polynomials are:
  FP2(z; LHCnr = 1) = 1
  FP2(z; LHCnr = 2) = (1+z)
  FP2(z; LHCnr = 3) = 1+8*z-11*z^2-6*z^3
Some GF2(z;LHCnr) are:
  GF2(z; LHCnr = 3) = (1+8*z-11*z^2-6*z^3)/((1-z)^3*(1-2*z)^2*(1-3*z))
  GF2(z; LHCnr = 4) = (1+38*z-108*z^2-242*z^3+839*z^4-444*z^5-180*z^6)/((1-z)^4*(1-2*z)^3*(1-3*z)^2*(1-4*z))

Cf. A156920, A156921, A156927, A156933.
For the first few GF2's see A050488, A142965, A142966 and A142968.
Row sums(n) = A156926(n).
The number of FP2 terms follow the 'Lazy Caterer's sequence' A000124.
For the polynomials in the denominators of the GF2(z;LHCnr) see A157703.

LHCnr:=5; LHCmax:=(LHCnr)*(LHCnr-1)/2: RHCend:=LHCnr+LHCmax: for k from LHCnr 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); 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..RHCend); for nx from 0 to LHCmax do num:=sort(sum(A[t]*z^t,t=0..LHCmax)): nom:=product((1-u*z)^(LHCnr-u+1),u=1..LHCnr); LHCb:=series(num/nom,z,nx+1); y:=coeff(LHCb,z,nx)-A[nx]; x:=LHC[LHCnr+nx]; A[nx]:=x-y; end do: FP2[LHCnr]:=sort(num,z, ascending); GenFun[LHCnr]:= FP2[LHCnr]/ product((1-m*z)^(LHCnr-m+1), m=1..LHCnr);

G.f.: GF2(z; LHCnr) = FP2(z; LHCnr)/Product_{m=1..LHCnr} (1-m*z)^(LHCnr-m+1).

Row sum(n+1) = (-1)^(n)*2*(n+1)!*Row sum(n); Row sum(n=0) = 1.

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).

A156933 FP4 polynomials related to the o.g.f.s of the columns of the A156925 matrix.

Original entry on oeis.org

1, 1, 1, -11, 156, -627, 736, 591, -1116, -369, -6, 106, -2772, 76070, -1087552, 8632650, -40358780, 106452214, -99774996, -284430514, 1125952500, -1581820542, 737716032, 414532350, -357790500, -81870750, -1275750

Offset: 0

Johannes W. Meijer, Feb 20 2009

For the matrix of the coefficients of the FP2 see A156925. The coefficients in the columns of this matrix are the powers of z^m, m=0, 1, 2, ... . The columns are numbered: 1, 2, 3, ... .
The GF4(z;m) generate the sequences of the z^m coefficients. The general structure of the GF4(z;m) is given below.
The FP4(z,m) in the numerator of the GF4(z;m) is a polynomial of a certain degree, let's say k4. The (k4+1) coefficients of this polynomial can be determined one by one by comparing the series expansion of the FP4(z,m) with the values of the powers of z^m in column (m+1). These values can be generated with the GF2 formulas, see A156925.
An appropriate name for the polynomials FP4(z;m) in the numerators of the GF4(z;m) seems to be the flower polynomials of the fourth kind because the zero patterns of these polynomials look like flowers. The zero patterns of the FP4 and the FP3, see A156927, resemble each other closely and look like the zero patterns of the FP1 and FP2.
The sequence of (k4+1) number of terms of the FP4(z;m) polynomials for m from 0 to 11 is 1, 2, 7, 17, 28, 44, 63, 83, 108, 136, 167, 199.

			The first few rows of the "triangle" of the FP4(z;m) coefficients are:
[1]
[1, 1]
[ -11, 156, -627, 736, 591, -1116, -369]
The first few FP4 polynomials are:
FP4(z; m=0) = 1
FP4(z; m=1) = (1+z)
FP4(z; m=2) = ( -11+156*z-627*z^2+736*z^3+591*z^4-1116*z^5-369*z^6 )
Some GF4(z;m) are:
GF4(z;m=1) = z*(1+z)/((1-3*z)*(1-z)^4)
GF4(z;m=2) = z^2*(-11+156*z-627*z^2+736*z^3+591*z^4-1116*z^5-369*z^6)/((1-z)^7*(1-3*z)^4*(1-5*z))

Cf. A156920, A156921, A156925, A156927.
For the first few GF4's see A156934, A156935, A156936, A156937.
Row sums A156938.
For the polynomials in the denominators of the GF4(z;m) see A157705. - Johannes W. Meijer, Mar 07 2009

G.f.: GF4(z;m):= z^q*FP4(z;m) / Product_{k=0..m} (1-(2*m+1-(2*k))*z)^(3*k+1).

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A157704 G.f.s of the z^p coefficients of the polynomials in the GF3 denominators of A156927.

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A156932 Row sums of the FP3 polynomials of A156927.

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A157707 The z^2 coefficients of the polynomials in the GF3 denominators of A156927 divided by 2.

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A002414 Octagonal pyramidal numbers: a(n) = n*(n+1)*(2*n-1)/2.

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

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A156925 FP2 polynomials related to the generating functions of the left hand columns of the A156920 triangle.

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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)!.

A002414 Octagonal pyramidal numbers: a(n) = n(n+1)(2*n-1)/2.

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)!.