A086915 → θ+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	4	8														12
2	12	48	24													84
3	48	288	288	64												688
4	240	1920	2880	1280	160											6480
5	1440	14400	28800	19200	4800	384										69024
6	10080	120960	302400	268800	100800	16128	896									820064
7	80640	1128960	3386880	3763200	1881600	451584	50176	2048								10745088
8	725760	11612160	40642560	54190080	33868800	10838016	1806336	147456	4608							153835776
9	7257600	130636800	522547200	812851200	609638400	243855360	54190080	6635520	414720	10240						2388037120
10	79833600	1596672000	7185024000	12773376000	11176704000	5364817920	1490227200	243302400	22809600	1126400	22528					39933915648
11	958003200	21076070400	105380352000	210760704000	210760704000	118025994240	39341998080	8028979200	1003622400	74342400	2973696	49152				715413792768
12	12454041600	298896998400	1643933491200	3653185536000	4109833728000	2630293585920	1022891950080	250504151040	39141273600	3865804800	231948288	7667712	106496			13665240283136
13	174356582400	4533271142400	27199626854400	66487976755200	83109970944000	59839179079680	26595190702080	7598625914880	1424742359040	175894118400	14071529472	697761792	19382272	229376		277153623355392
14	2615348736000	73229764608000	475993469952000	1269315919872000	1745309389824000	1396247511859200	698123755929600	227958777446400	49865982566400	7387552972800	738755297280	48843325440	2035138560	48168960	491520	5946837156188160

Technical details

Node id: n-b8be27de1bfb545f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9678f4e916f31c27

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:       4,         8;
  2:      12,        48,        24;
  3:      48,       288,       288,        64;
  4:     240,      1920,      2880,      1280,       160;
  5:    1440,     14400,     28800,     19200,      4800,       384;
  6:   10080,    120960,    302400,    268800,    100800,     16128,      896;
  7:   80640,   1128960,   3386880,   3763200,   1881600,    451584,    50176,    2048;
  8:  725760,  11612160,  40642560,  54190080,  33868800,  10838016,  1806336,  147456,   4608;
  9: 7257600, 130636800, 522547200, 812851200, 609638400, 243855360, 54190080, 6635520, 414720, 10240;

%F Formula

The row polynomials satisfy n*P(n,x) = (n^2+n+(2*n+2)*x)*P(n-1,x) + (n+1)*x*d/dx 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[((n^2 + n + (2*n + 2)*x)*P[n - 1, x] + (n + 1)*x*D[P[n - 1, x], x])/(n)]];
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] = ((n^2 + n + (2*n + 2)*x)*v[i - 1] + (n + 1)*x*deriv(v[i - 1], x))/(n)); 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 (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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

Related ?

Same fingerprint ?

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.