A124848 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	1														5
2	10	8	1													19
3	20	30	12	1												63
4	35	80	60	16	1											192
5	56	175	200	100	20	1										552
6	84	336	525	400	150	24	1									1520
7	120	588	1176	1225	700	210	28	1								4048
8	165	960	2352	3136	2450	1120	280	32	1							10496
9	220	1485	4320	7056	7056	4410	1680	360	36	1						26624
10	286	2200	7425	14400	17640	14112	7350	2400	450	40	1					66304
11	364	3146	12100	27225	39600	38808	25872	11550	3300	550	44	1				162560
12	455	4368	18876	48400	81675	95040	77616	44352	17325	4400	660	48	1			393216
13	560	5915	28392	81796	157300	212355	205920	144144	72072	25025	5720	780	52	1		940032
14	680	7840	41405	132496	286286	440440	495495	411840	252252	112112	35035	7280	910	56	1	2224128

Technical details

Node id: n-bdca8d2826fff8ca · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-f91847f627145760

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:   4,    1;
  2:  10,    8,    1;
  3:  20,   30,   12,    1;
  4:  35,   80,   60,   16,    1;
  5:  56,  175,  200,  100,   20,    1;
  6:  84,  336,  525,  400,  150,   24,    1;
  7: 120,  588, 1176, 1225,  700,  210,   28,   1;
  8: 165,  960, 2352, 3136, 2450, 1120,  280,  32,  1;
  9: 220, 1485, 4320, 7056, 7056, 4410, 1680, 360, 36, 1;

%F Formula

The row polynomials satisfy n*P(n,x) = (n+3+(2*n-1)*x)*P(n-1,x) - (x+x^2)*d/dx 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[Together[((n + 3 + (2*n - 1)*x)*P[n - 1, x] - (x + x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + 3 + (2*n - 1)*x)*v[i - 1] - (x + x^2)*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 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: T(n,k) = (k+1)*(k+2)*(k+3)*binomial(n,k)/6 (0 <= k <= n).?
  - 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 ?

A124848: Triangle read by rows: T(n,k) = (k+1)*(k+2)*(k+3)*binomial(n,k)/6 (0 <= k <= n).

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.