A151775 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-1}(x) + \left(24 + 32\,x + 10\,x^{2}\right)\,P_{n-1}'(x) + \left(16 + 32\,x + 20\,x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	57	76	24													157
3	2763	5462	3480	720												12425
4	250737	655576	627984	262080	40320											1836697
5	36581523	118940522	151991760	95588640	29635200	3628800										436366445
6	7828053417	30435667876	48615763944	40879776960	19105511040	4710182400	479001600									152053957237
7	2309644635483	10450328632382	20027633993640	21089047739760	13187071497600	4900146451200	1002550348800	87178291200								73053601590065
8	898621108880097	4638024381708976	10367787111743904	13117171658048640	10278725669372160	5111018283571200	1575660435148800	275483400192000	20922789888000							46283414838553777
9	445777636063460643	2584568731265561042	6601458159781401120	9753008378506390080	9188659740573331200	5727242569227379200	2362582180204646400	622221802339737600	94968543301632000	6402373705728000						37386890114969267285
10	274613643571568682777	1767003579098551930876	5076387729987563223864	8577226877314301781120	9441905032060102183680	7077840721244847590400	3660140032336653465600	1289693744313182208000	296428835512922112000	40142883134914560000	2432902008176640000					37503815980582784378317
11	205676334188681975553003	1454363558536372507006502	4641611191824367177606200	8827727764218659263875600	11119297635691576642080000	9742064570824382259724800	6059703478933069097932800	2676608122096495390003200	822965307199242375168000	167785208639002386432000	20419346554626539520000	1124000727777607680000				45739346519434253222582105
12	184053312545818735778213457	1418636396066470814920285576	4979530036034152519030459824	10527244243914376505427278400	14932238921352223537387305600	14974112439377322296212224000	10887825184961030588975616000	5784858629278206778460160000	2229477692739862359674880000	607950182688404690534400000	111361870772203517706240000	12305559967709248880640000	620448401733239439360000			66650214918099514832427062857
13	193944394596325636374396208563	1618343191939023342646196648762	6196015703901474285926466800880	14414043053348083462993306132320	22735867860188939644504203888000	25681910809363065296486647200000	21376722124205391184636082688000	13277670693955683324901143552000	6155211363601792706465771520000	2103688762564469654648954880000	515316947013254249953689600000	85689353202875532420710400000	8670766414222021165056000000	403291461126605635584000000		114363498315755726948758209518525
14	237694572315172714282278845824137	2134731792465439600736487345525076	8852987793860896474680926272907784	22473983931687146431211073017244480	39021272221852106675993856348097920	49027639824549381392099164478016000	45975413786627410063165783499904000	32693226856916321075056973936640000	17718029522317785789841252700160000	7283582061579659737291828838400000	2235980779470155164798852300800000	497141840501412215976670003200000	75685990813652868253994188800000	7063246650171371101618176000000	304888344611713860501504000000	228234739109951323288351261455519397

Technical details

Node id: n-460e179a989db7f3 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-2285acaa57b31e0b

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:                  1;
  1:                  3,                   2;
  2:                 57,                  76,                  24;
  3:               2763,                5462,                3480,                 720;
  4:             250737,              655576,              627984,              262080,               40320;
  5:           36581523,           118940522,           151991760,            95588640,            29635200,             3628800;
  6:         7828053417,         30435667876,         48615763944,         40879776960,         19105511040,          4710182400,           479001600;
  7:      2309644635483,      10450328632382,      20027633993640,      21089047739760,      13187071497600,       4900146451200,       1002550348800,        87178291200;
  8:    898621108880097,    4638024381708976,   10367787111743904,   13117171658048640,   10278725669372160,    5111018283571200,    1575660435148800,    275483400192000,    20922789888000;
  9: 445777636063460643, 2584568731265561042, 6601458159781401120, 9753008378506390080, 9188659740573331200, 5727242569227379200, 2362582180204646400, 622221802339737600, 94968543301632000, 6402373705728000;

%F Formula

The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) + (24+32*x+10*x^2)*d/dx P(n-1,x) + (16+32*x+20*x^2+4*x^3)*d^2/dx^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[(3 + 2*x)*P[n - 1, x] + (24 + 32*x + 10*x^2)*D[P[n - 1, x], x] + (16 + 32*x + 20*x^2 + 4*x^3)*D[P[n - 1, x], {x, 2}]];
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] = (3 + 2*x)*v[i - 1] + (24 + 32*x + 10*x^2)*deriv(v[i - 1], x) + (16 + 32*x + 20*x^2 + 4*x^3)*deriv(deriv(v[i - 1], x), 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]
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 Triangle read by rows: T(n,k) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.