A008955 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} + n^{2} + 5\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	5														7
2	10	59	85													154
3	100	960	3033	3145												7238
4	1700	22820	113961	250610	204425											593516
5	44200	765020	5267806	18025921	30626660	20646925										75376532
6	1635400	34714740	305836722	1430790947	3746944965	5204801925	2993804125									13718528824
7	81770000	2057910800	22130639880	131789381584	469213064809	998388254355	1175036185475	589779412625								3388476619528
8	5315050000	154779092000	1967374667800	14253884252120	64368720279673	185482994188988	332963133425110	340319961487700	151573309044625							1091089471488016
9	435834100000	14419276794000	211627927659600	1808215275708840	9910747444872186	36129439614390741	87584950052280120	136119255205152150	123032998825161750	49261325439503125						444073414895622512
10	44019244100000	1631116430294000	27156550688013600	267492541838092440	1726079817491335626	7623283126447211427	23333985240650979261	48869609746684695270	67010154218607348900	54311626398279677375	19753791501240753125					222924854277602501024
11	5370347780200000	220345537884368000	4104190652630249200	45805017187933873680	340315620525417779772	1767189252909857572704	6544038515686317011937	17279075230811257764525	31876999541812173771750	39125943216614684856250	28751101366317015408125	9580588878101765265625				135315386546504718121568
12	778700428129000000	35048793662408760000	722247019990666470000	9009845498818065472000	75775259893623423180300	452604554715095406970524	1968553783703503786181073	6281376132018637291743774	14592191341740860927034675	24066290502034753570456000	26744578934102640396234375	17978570875689673854003750	5527999782664718558265625			97697709086576406483772096
13	132379072781930000000	6485475112452822200000	146510026707864030420000	2020634967332752330430000	18981459584615812265195000	128242625249549276678052180	641067426771715234169827158	2400544854010440402841028001	6733164169472563804106429748	13970182923704470954579994975	20839457088674977034558555750	21162436987254732103431309375	13111252445894911354065695000	3742455852864014463945828125		82749959596330216291754965312
14	26078677338040210000000	1381556169287021023400000	33953573224724679419740000	513075459530225472974410000	5325545987525525595630965000	40164242948084620133754354460	226960743894924083323726911426	976145266255853218182996835227	3210861051784290785639173641141	8037659909005743364275806362255	15071966641576980175153331538125	20527973901099039196504434210625	19195429766836262237944519774375	11029596973041716262338898715625	2937827844498251354197475078125	81260460323596272398503375936384

Technical details

Node id: n-8378a9feb3237cd5 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-f6632b3a6a00b47e

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:            1;
  1:            2,              5;
  2:           10,             59,              85;
  3:          100,            960,            3033,             3145;
  4:         1700,          22820,          113961,           250610,           204425;
  5:        44200,         765020,         5267806,         18025921,         30626660,          20646925;
  6:      1635400,       34714740,       305836722,       1430790947,       3746944965,        5204801925,        2993804125;
  7:     81770000,     2057910800,     22130639880,     131789381584,     469213064809,      998388254355,     1175036185475,       589779412625;
  8:   5315050000,   154779092000,   1967374667800,   14253884252120,   64368720279673,   185482994188988,   332963133425110,    340319961487700,    151573309044625;
  9: 435834100000, 14419276794000, 211627927659600, 1808215275708840, 9910747444872186, 36129439614390741, 87584950052280120, 136119255205152150, 123032998825161750, 49261325439503125;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (n^2+1+(5*n^2+2)*x+(4*n^2+1)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 1 + (5*n^2 + 2)*x + (4*n^2 + 1)*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 1 + (5*n^2 + 2)*x + (4*n^2 + 1)*x^2)*v[i - 1])/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.