A286785 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(56 + 30\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(240 + 84\,x + 162\,n + 66\,n x + 27\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(-30\,x - 28\,x^{2} - 9\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 30 + 18\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	30	18	2															50
1	143	132	33	2														310
2	728	910	364	52	2													2056
3	3876	6120	3400	800	75	2												14273
4	21318	40698	29070	9690	1530	102	2											102410
5	120175	269192	235543	102410	23275	2660	133	2										753390
6	690690	1776060	1841840	991760	297528	49588	4312	168	2									5651948
7	4032015	11705850	14047020	9041760	3390660	753480	96600	6624	207	2								43074218
8	23841480	77134200	105183000	78887250	35626500	9975420	1719900	175500	9750	250	2							332553252
9	142498692	508373712	776682060	665727480	352443960	119617344	26166294	3617460	301455	13860	297	2						2595442616
10	859515920	3352112088	5672805072	5473759280	3328637400	1331454960	355054656	62656704	7120080	494450	19140	348	2					20443630100
11	5225264024	22116233776	41073005584	44078347456	30303863876	13986398712	4416757488	954974592	139267128	13263536	780208	25792	403	2				162308182577
12	31983672534	146012418090	295269556582	348954930506	267802621086	140277563426	51321059790	13196843946	2368664298	290888598	23585562	1191190	34034	462	2			1297503030106
13	196947587823	964641246480	2110152726675	2723743235850	2309260569525	1354766200788	564485916995	168783164550	36167820975	5488883400	576332757	40302990	1767675	44100	525	2		10435055801110
14	1219199353190	6377350462840	15005530500800	21007742701120	19507189651040	12679673273176	5935166212976	2027541252880	506885313220	92160966040	12002358368	1091123488	66531920	2558920	56240	592	2	84371602316812

Technical details

Node id: n-664dba754f71e70d · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-364ead120df7d525

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:        30,        18,         2;
  1:       143,       132,        33,         2;
  2:       728,       910,       364,        52,         2;
  3:      3876,      6120,      3400,       800,        75,         2;
  4:     21318,     40698,     29070,      9690,      1530,       102,        2;
  5:    120175,    269192,    235543,    102410,     23275,      2660,      133,       2;
  6:    690690,   1776060,   1841840,    991760,    297528,     49588,     4312,     168,      2;
  7:   4032015,  11705850,  14047020,   9041760,   3390660,    753480,    96600,    6624,    207,     2;
  8:  23841480,  77134200, 105183000,  78887250,  35626500,   9975420,  1719900,  175500,   9750,   250,   2;
  9: 142498692, 508373712, 776682060, 665727480, 352443960, 119617344, 26166294, 3617460, 301455, 13860, 297, 2;

%F Formula

The row polynomials satisfy (4*n^2+30*n+56)*P(n,x) = (27*n^2+162*n+240+(12*n^2+66*n+84)*x)*P(n-1,x) - ((9*n+30)*x+(8*n+28)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 30 + 18*x + 2*x^2.

%t Mathematica

P[0, x_] := 30 + 18*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^2 + 162*n + 240 + (12*n^2 + 66*n + 84)*x)*P[n - 1, x] - ((9*n + 30)*x + (8*n + 28)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 30*n + 56)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 30 + 18*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((27*n^2 + 162*n + 240 + (12*n^2 + 66*n + 84)*x)*v[i - 1] - ((9*n + 30)*x + (8*n + 28)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 30*n + 56)); 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 T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.?
  - 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

[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.