A292219 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(7 + x\right)\,P_{n-1}(x) + \left(32 + 8\,x\right)\,P_{n-1}'(x) + \left(16 + 16\,x\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	7	1														8
3	81	22	1													104
4	1303	507	45	1												1856
5	26785	13324	1734	76	1											41920
6	669351	400405	67990	4410	115	1										1142272
7	19674097	13625826	2832975	241820	9375	162	1									36384256
8	664400311	519241807	127013691	13381235	685685	17661	217	1								1324740608
9	25330978113	21930940312	6141405340	766902696	48704390	1662248	30492	280	1							54220623872
10	1075631907655	1017492654249	319779823644	46032713076	3478889106	147508830	3591084	49284	351	1						2462567137280
11	50322142646161	51456025246510	17878656217005	2907241268520	254237346930	12858131412	389491410	7101480	75645	430	1					122831557525504
12	2570964805355607	2817608679954211	1069669012404025	193473688610565	19181356753350	1125079945374	40710862962	924523050	13092915	111375	517	1				6672064271613952
13	142389639792952801	166096794302069412	68249764675118466	13565413102923220	1501126189992495	100039093585992	4204281312924	114167157192	2015244495	22803220	158466	612	1			391907097643319296
14	8495817365820681703	10488098621015995837	4628616926069611410	1001150461610854126	122146154464110205	9104960960452071	435373997708364	13744643183700	290314344177	4096676155	37884418	219102	715	1		24745383903031721984
15	543205619067484203825	706274965898385310714	332619238207010209819	77664638171866017636	10344211265628160921	851887046284824550	45625852779389883	1640448013703928	40310728845387	681013955622	7856874025	60486244	295659	826	1	1671007866957112279040

Technical details

Node id: n-ed9c4455f45850cc · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-68a0e237a0942a37

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:             7,             1;
   3:            81,            22,            1;
   4:          1303,           507,           45,           1;
   5:         26785,         13324,         1734,          76,          1;
   6:        669351,        400405,        67990,        4410,        115,         1;
   7:      19674097,      13625826,      2832975,      241820,       9375,       162,       1;
   8:     664400311,     519241807,    127013691,    13381235,     685685,     17661,     217,     1;
   9:   25330978113,   21930940312,   6141405340,   766902696,   48704390,   1662248,   30492,   280,   1;
  10: 1075631907655, 1017492654249, 319779823644, 46032713076, 3478889106, 147508830, 3591084, 49284, 351, 1;

%F Formula

The row polynomials satisfy P(n,x) = (7+x)*P(n-1,x) + (32+8*x)*d/dx P(n-1,x) + (16+16*x)*d^2/dx^2 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[(7 + x)*P[n - 1, x] + (32 + 8*x)*D[P[n - 1, x], x] + (16 + 16*x)*D[P[n - 1, x], {x, 2}]];
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] = (7 + x)*v[i - 1] + (32 + 8*x)*deriv(v[i - 1], x) + (16 + 16*x)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 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. A generalization of unsigned Lah numbers, called L[4,3].?
  - 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.