A089503 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	1														5
3	30	12	1													43
4	336	168	24	1												529
5	5040	2880	540	40	1											8501
6	95040	59400	13200	1320	60	1										169021
7	2162160	1441440	360360	43680	2730	84	1									4010455
8	57657600	40360320	11007360	1528800	117600	5040	112	1								110676833
9	1764322560	1283143680	374250240	57576960	5140800	274176	8568	144	1							3484717129
10	60949324800	45711993600	14065228800	2344204800	234420480	14651280	574560	13680	180	1						123320412181
11	2346549004800	1805037696000	580190688000	103145011200	11281485600	796340160	36867600	1108800	20790	220	1					4847038223171
12	99638080819200	78287063500800	26095687833600	4892941468800	575640172800	44772013440	2356421760	84157920	2003760	30360	264	1				209536628422705
13	4626053752320000	3700843001856000	1272164781888000	249444074880000	31180509360000	2625727104000	153167414400	6251731200	177606000	3432000	42900	312	1			9882471447634813
14	233153109116928000	189436901157504000	66860082761472000	13619646488448000	1792058748480000	161285287363200	10240335705600	465469804800	15178363200	351351000	5621616	58968	364	1		505033804901100749
15	12677700308232960000	10440459077368320000	3770165777938560000	793719111144960000	109136377782432000	10393940741184000	708677777808000	35213802624000	1283836554000	34235641440	658377720	8867040	79170	420	1	27802319803528367791

Technical details

Node id: n-2e9b77d16b34b95b · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-fe953b9def4ff3da

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:           4,           1;
   3:          30,          12,           1;
   4:         336,         168,          24,          1;
   5:        5040,        2880,         540,         40,         1;
   6:       95040,       59400,       13200,       1320,        60,        1;
   7:     2162160,     1441440,      360360,      43680,      2730,       84,      1;
   8:    57657600,    40360320,    11007360,    1528800,    117600,     5040,    112,     1;
   9:  1764322560,  1283143680,   374250240,   57576960,   5140800,   274176,   8568,   144,   1;
  10: 60949324800, 45711993600, 14065228800, 2344204800, 234420480, 14651280, 574560, 13680, 180, 1;

%F Formula

The row polynomials satisfy P(n,x) = (3*n-2+x)*P(n-1,x) + (n-1+x)*d/dx 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[(3*n - 2 + x)*P[n - 1, x] + (n - 1 + x)*D[P[n - 1, x], x]];
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] = (3*n - 2 + x)*v[i - 1] + (n - 1 + x)*deriv(v[i - 1], 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 (−∞, 0]
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 of numbers used for basis change between certain falling factorials.?
  - 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 ?

A089503: Triangle of numbers used for basis change between certain falling factorials.

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.