A086915 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2\right)\,P_{n}(x) = \left(8 + 4\,x\right)\,P_{n-1}(x) + \left(6 + 4\,x\right)\,P_{n-1}'(x) + \left(1 + x\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	8	4														12
2	44	40	8													92
3	304	384	144	16												848
4	2512	3968	2112	448	32											9072
5	24064	44960	30400	9280	1280	64										110048
6	261536	558528	452160	177280	35520	3456	128									1488608
7	3173888	7571200	7074816	3351040	869120	123648	8960	256								22172928
8	42483968	111351808	117168128	64540672	20357120	3770368	401408	22528	512							360096512
9	621159424	1766877696	2056679424	1283874816	473647104	107218944	14923776	1234944	55296	1024						6325672448
10	9841950208	30091965440	38235571200	26549944320	11155845120	2966777856	506634240	54988800	3640320	133120	2048					119407452672
11	167879268352	547534524416	751602903040	572443576320	268738990080	81720274944	16451162112	2197155840	191262720	10362880	315392	4096				2408769800192
12	3065723549696	10599516733440	15589649129472	12882146140160	6659529338880	2270590451712	524295045120	82769854464	8886620160	634388480	28655616	737280	8192			51683770652672
13	59651093528576	217502529826816	340459156217856	302578192277504	170307831316480	64147228434432	16655060877312	3023389704192	384528089088	33919508480	2021933056	77316096	1703936	16384		1174745030750208
14	1231571119812608	4715266036375552	7811463604625408	7413713483251712	4502356984373248	1851785053454336	532524584976384	108947699367936	15965257211904	1671182729216	123281686528	6229164032	204259328	3899392	32768	28185394725220352

Technical details

Node id: n-227b620650ca6bf3 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-7b312ef315523524

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:         2;
  1:         8,          4;
  2:        44,         40,          8;
  3:       304,        384,        144,         16;
  4:      2512,       3968,       2112,        448,        32;
  5:     24064,      44960,      30400,       9280,      1280,        64;
  6:    261536,     558528,     452160,     177280,     35520,      3456,      128;
  7:   3173888,    7571200,    7074816,    3351040,    869120,    123648,     8960,     256;
  8:  42483968,  111351808,  117168128,   64540672,  20357120,   3770368,   401408,   22528,   512;
  9: 621159424, 1766877696, 2056679424, 1283874816, 473647104, 107218944, 14923776, 1234944, 55296, 1024;

%F Formula

The row polynomials satisfy 2*P(n,x) = (8+4*x)*P(n-1,x) + (6+4*x)*d/dx P(n-1,x) + (1+x)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 2.

%t Mathematica

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((8 + 4*x)*P[n - 1, x] + (6 + 4*x)*D[P[n - 1, x], x] + (1 + x)*D[P[n - 1, x], {x, 2}])/(2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((8 + 4*x)*v[i - 1] + (6 + 4*x)*deriv(v[i - 1], x) + (1 + x)*deriv(deriv(v[i - 1], x), x))/(2)); 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) = 2^k * (n!/k!)*binomial(n-1,k-1).?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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.