A225473 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	3														7
3	32	40	9													81
4	384	672	316	27												1399
5	6144	13824	9920	2320	81											32289
6	122880	337920	326400	127680	16564	243										931687
7	2949120	9584640	11642880	6428160	1536992	116920	729									32259441
8	82575360	309657600	453304320	324065280	114866304	17842272	821356	2187								1303134679
9	2642411520	11230248960	19240058880	16907304960	7995101184	1937101824	202835840	5758240	6561							60160827969
10	95126814720	451852369920	886859366400	925256908800	548770037760	182900551680	31490557440	2277260160	40333924	19683						3124574220487
11	3805072588800	19976631091200	44194332672000	53399833804800	38152417443840	16279721902080	3978959400960	499685483520	25372533152	282416200	59049					180312309395601
12	167423193907200	962683364966400	2370084588748800	3255596325273600	2723662536376320	1420828596633600	454012791767040	83595196354560	7799752650624	281357194272	1977149596	177147				11445969681199159
13	8036313307545600	50226958172160000	136198768243507200	209634766671052800	201103072548618240	124216739333406720	49258679111516160	12114198137733120	1713104773146624	120372576090624	3110746333760	13840755760	531441			792626097462398049
14	417888291992371200	2820745970948505600	8353412836805836800	14241937701312921600	15415466511921315840	11013543197870653440	5229961934458060800	1620725091316531200	312888652630794240	34474951907235840	1842917597364480	34328935717440	96887416084	1594323		59462922484586318887
15	23401744351572787200	169662646548902707200	544821860685053952000	1019255662187642880000	1229021981633308262400	997690749088274841600	552832793828130816000	207509122525062758400	51268905128685404160	7885938048652984320	684510767795312640	28055711189076480	378393392220512	678218289880	4782969	4804064349575887075761

Technical details

Node id: n-c7c3b28456d49b6f · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-efc167c4cad578f4

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 -1)
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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:           1;
   2:           4,            3;
   3:          32,           40,            9;
   4:         384,          672,          316,           27;
   5:        6144,        13824,         9920,         2320,           81;
   6:      122880,       337920,       326400,       127680,        16564,          243;
   7:     2949120,      9584640,     11642880,      6428160,      1536992,       116920,         729;
   8:    82575360,    309657600,    453304320,    324065280,    114866304,     17842272,      821356,       2187;
   9:  2642411520,  11230248960,  19240058880,  16907304960,   7995101184,   1937101824,   202835840,    5758240,     6561;
  10: 95126814720, 451852369920, 886859366400, 925256908800, 548770037760, 182900551680, 31490557440, 2277260160, 40333924, 19683;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n - 4 + (4*n - 5)*x)*P[n - 1, x] - (4*x + 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (4*n - 4 + (4*n - 5)*x)*v[i - 1] - (4*x + 4*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A225473: Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >=…

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.