A330798 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(\left(2\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(4 + 3\,x - 20\,n - 12\,n x + 8\,n^{2} + 3\,n^{2} x + 32\,n^{3} + 18\,n^{3} x\right)\,P_{n-1}(x) + \left(-8 - 10\,x - 3\,x^{2} + 12\,n + 18\,n x + 6\,n x^{2} + 20\,n^{2} + 28\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

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	2														6
2	30	33	9													72
3	264	452	256	48												1020
4	2490	5795	5025	1925	275											15510
5	24396	71814	84096	48984	14196	1638										245124
6	244790	871605	1286985	1008910	442960	103292	9996									3968538
7	2496528	10430216	18597888	18349120	10820160	3813984	744192	62016								65314104
8	25763058	123542919	258229377	307321371	227795085	107697681	31719303	5321349	389367							1087779510
9	268243860	1451989130	3481484800	4853668400	4336255000	2574715220	1016128960	257048000	37823500	2466750						18279823620
10	2812481870	16961031241	45889110729	73356017196	76731451104	54881483580	27184177020	9208279080	2041580970	267543705	15737865					309348894360
11	29653804824	197149648332	594130440960	1071362502000	1284522876000	1075254944400	641273068800	272495901600	80855868600	15956261100	1884879360	100975680				5264641171656
12	314097641130	2282287814443	7581308383649	15224432928705	20585879369055	19746173785338	13778154275886	7046817758838	2621976580080	692194597380	123076872204	13234405548	650872404			90010285284660
13	3339741725404	26330547468302	95583230391936	211547561956808	318496408487180	344470721576058	275360773949184	164727330442752	73749604941936	24406504125080	5803443666944	938971381152	92655816176	4211628008		1544851707556920
14	35626286189670	302890060737885	1192947562348425	2885066193615450	4786622796727200	5763429037234440	5193852245473320	3558844668693600	1863241024015800	741768488389500	221045495462220	47814411233160	7097230420800	647082114600	27341497800	26600919924153870

Technical details

Node id: n-6fe98fbfbcdc3b9a · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-a09e6eb68394f014

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,          2;
  2:        30,         33,          9;
  3:       264,        452,        256,         48;
  4:      2490,       5795,       5025,       1925,        275;
  5:     24396,      71814,      84096,      48984,      14196,       1638;
  6:    244790,     871605,    1286985,    1008910,     442960,     103292,       9996;
  7:   2496528,   10430216,   18597888,   18349120,   10820160,    3813984,     744192,     62016;
  8:  25763058,  123542919,  258229377,  307321371,  227795085,  107697681,   31719303,   5321349,   389367;
  9: 268243860, 1451989130, 3481484800, 4853668400, 4336255000, 2574715220, 1016128960, 257048000, 37823500, 2466750;

%F Formula

The row polynomials satisfy (4*n^3+2*n^2)*P(n,x) = (32*n^3+8*n^2-20*n+4+(18*n^3+3*n^2-12*n+3)*x)*P(n-1,x) + (20*n^2+12*n-8+(28*n^2+18*n-10)*x+(9*n^2+6*n-3)*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[((32*n^3 + 8*n^2 - 20*n + 4 + (18*n^3 + 3*n^2 - 12*n + 3)*x)*P[n - 1, x] + (20*n^2 + 12*n - 8 + (28*n^2 + 18*n - 10)*x + (9*n^2 + 6*n - 3)*x^2)*D[P[n - 1, x], x])/(4*n^3 + 2*n^2)]];
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] = ((32*n^3 + 8*n^2 - 20*n + 4 + (18*n^3 + 3*n^2 - 12*n + 3)*x)*v[i - 1] + (20*n^2 + 12*n - 8 + (28*n^2 + 18*n - 10)*x + (9*n^2 + 6*n - 3)*x^2)*deriv(v[i - 1], x))/(4*n^3 + 2*n^2)); 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 (−∞, -1]
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, interpolating between the central binomial coefficients and the central coefficients of the Cata…?
  - 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

[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.