A269947 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-64 + 16\,x - x^{2} + 192\,n - 24\,n x - 240\,n^{2} + 12\,n^{2} x + 160\,n^{3} - 2\,n^{3} x - 60\,n^{4} + 12\,n^{5} - n^{6}\right)\,P_{n-2}(x) + \left(-9 + 2\,x + 15\,n - 9\,n^{2} + 2\,n^{3}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	2														2
2	0	2	3													5
3	0	16	27	4												47
4	0	432	753	144	5											1334
5	0	27648	48840	10220	500	6										87214
6	0	3456000	6146472	1342620	75275	1350	7									11021724
7	0	746496000	1332821952	298201216	17937675	381930	3087	8								2395841868
8	0	256048128000	458277673536	104060113024	6525374045	152527200	1504426	6272	9							825065326512
9	0	131096641536000	235022241042432	53889814766336	3471066652320	85924375254	948214512	4930608	11664	10						423566641529136
10	0	95569451679744000	171527858682236928	39599037952715520	2597769857999200	66804149542950	791493483711	4678086960	14049990	20250	11					309361718507879520
11	0	95569451679744000000	171671212859756544000	39827741764291835904	2647268655440094400	69921473372549040	869431658177775	5582650941384	19312837830	35861100	33275	12				309786471466262874720
12	0	127202940185739264000000	228637738493855576064000	53239619238752108980224	3573299257596130441280	96242203445390885520	1238788589302592405	8424144583756704	31985869460787	69189832800	83736235	52272	13			412751086624177499802240
13	0	219806680640957448192000000	395276816527661044334592000	92302912362555451752579072	6241210641174553538757120	170594486662750806708096	2252909253001169042280	15972680228505975832	64748745084966228	155099886034830	220805030160	181674636	79092	14		713800483605848544257439360
14	0	482915277368183513677824000000	868752875932232750575386624000	203316534215904542226195677184	13827318419113688439340116480	382285539967472986584195456	5148668529950110993715272	37666731894028964951224	160222258208740025727	412697499712928430	655718525899833	640017390012	370579313	115934	15	1569199477971299424112656218880

Technical details

Node id: n-42247a28dbf8f7f6 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-b56e8fc38add20bd

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: 0,               2;
  2: 0,               2,               3;
  3: 0,              16,              27,              4;
  4: 0,             432,             753,            144,             5;
  5: 0,           27648,           48840,          10220,           500,           6;
  6: 0,         3456000,         6146472,        1342620,         75275,        1350,         7;
  7: 0,       746496000,      1332821952,      298201216,      17937675,      381930,      3087,       8;
  8: 0,    256048128000,    458277673536,   104060113024,    6525374045,   152527200,   1504426,    6272,     9;
  9: 0, 131096641536000, 235022241042432, 53889814766336, 3471066652320, 85924375254, 948214512, 4930608, 11664, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(n^6-12*n^5+60*n^4-160*n^3+240*n^2-192*n+64+(2*n^3-12*n^2+24*n-16)*x+x^2)*P(n-2,x) + (2*n^3-9*n^2+15*n-9+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (2*n^3 - 12*n^2 + 24*n - 16)*x + x^2)*P[n - 2, x] + (2*n^3 - 9*n^2 + 15*n - 9 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (2*n^3 - 12*n^2 + 24*n - 16)*x + x^2)*v[i - 2] + (2*n^3 - 9*n^2 + 15*n - 9 + 2*x)*v[i - 1]); 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, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Parent-row provers of polytool 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.