A269947 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 - x + 3\,n x - 3\,n^{2} x + n^{3} x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	1														1
2	1	1													2
3	1	9	8												18
4	1	36	251	216											504
5	1	100	2555	16280	13824										32760
6	1	225	15055	335655	2048824	1728000									4127760
7	1	441	63655	3587535	74550304	444273984	373248000								895723920
8	1	784	214918	25421200	1305074809	26015028256	152759224512	128024064000							308129028480
9	1	1296	616326	135459216	14320729209	694213330464	13472453691584	78340747014144	65548320768000						158070191610240
10	1	2025	1561110	584760870	113070497673	11134024923825	519553971599840	9899759488178880	57175952894078976	47784725839872000					115391239875475200
11	1	3025	3586110	2145870870	697831367673	124204522596825	11653578895424840	529453731088018880	9956935441072958976	57223737619918848000	47784725839872000000				115506631115350675200
12	1	4356	7612385	6918983280	3553985495643	1053018072969588	176969798471798915	16040367240898480920	714659851519226088256	13309904809688027245056	76212579497951858688000	63601470092869632000000			153854832645647099366400
13	1	6084	15139553	20073184560	15509988603483	7194305009440692	1996585028563246979	321844179000167006040	28432414443791801118016	1248242128234910707751424	23075728090638862938144768	131758938842553681444864000	109903340320478724096000000		266015005644323834804505600
14	1	8281	28506101	53334782501	59610775081803	41269749971292843	17802473134304447303	4708341486753620618903	735524075707158713387896	63714256661245497764032576	2765463683822737687868023296	50829133553976135556548919296	289584291977410916858462208000	241457638684091756838912000000	584700982406223788900303308800

Technical details

Node id: n-681d69a78cec43b0 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-53af44bcc1c9acce

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 1,    1;
  3: 1,    9,      8;
  4: 1,   36,    251,       216;
  5: 1,  100,   2555,     16280,       13824;
  6: 1,  225,  15055,    335655,     2048824,      1728000;
  7: 1,  441,  63655,   3587535,    74550304,    444273984,      373248000;
  8: 1,  784, 214918,  25421200,  1305074809,  26015028256,   152759224512,   128024064000;
  9: 1, 1296, 616326, 135459216, 14320729209, 694213330464, 13472453691584, 78340747014144, 65548320768000;

%F Formula

The row polynomials satisfy P(n,x) = (1+(n^3-3*n^2+3*n-1)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (n^3 - 3*n^2 + 3*n - 1)*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; for(n = 1, N, my(i = n + 1); v[i] = (1 + (n^3 - 3*n^2 + 3*n - 1)*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 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, 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

Related ?

Same fingerprint ?

A269947: 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,…

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.