A144442 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 5\,x\right)\,P_{n-2}(x) + \left(-3 - 4\,x + 5\,n + 5\,n x\right)\,P_{n-1}(x) + \left(-5\,x - 5\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	19	19	1													39
3	238	357	121	1												717
4	4141	8282	4874	733	1											18031
5	92292	230730	193390	59355	4405	1										580173
6	2512589	7537767	8198740	3834535	687410	26437	1									22797479
7	80864308	283025078	378474822	238624360	69920280	7768599	158629	1								1058836077
8	3004542341	12018169364	19071280147	15150247667	6222366045	1215516903	86697182	951781	1							56769771431
9	126595099862	569677949379	1046242561308	1003351867476	532341215532	151721918691	20528272772	961124619	5710693	1						3451425720333
10	5964992405219	29824962026095	62223217940490	69943099605390	45583983779680	17215943229774	3541652000060	340424104595	10617104570	34264165	1					234648926460039
11	310812580570698	1709469193138839	3993403009254385	5149294593103440	3984317532158840	1881354280045093	526392118932759	80269348766330	5579025104920	117056552895	205584997	1				17641008943213197
12	17746142054555881	106476852327335286	275333111184869809	400627742923775590	359118462424043470	203952767754208486	72638357868731546	15471531145634269	1782254956922395	90722337157135	1289232497654	1233509989	1			1453239235443241511
13	1101814870285318112	7161796656854567728	20308539872955972204	32917206075857602114	33557519656929814280	22299437725352515917	9678765822945804228	2672553742201298634	441898660708139934	38998114235330665	1467614495175468	14191219969107	7401059941	1		130180013011442568333
14	73910327019389092909	517372289135723650363	1596365418644180072511	2852352753100672980347	3259779664771178074274	2483037644834468385174	1273774237816562609193	434397838821562337571	94866698777872819192	12356688578059310564	844369903208200866	23658578892474987	156161394629714	44406359653	1	12599081746187570997319

Technical details

Node id: n-21a70536fc2023cb · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-83f6dcff4e22986d

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:            2,            1;
  2:           19,           19,             1;
  3:          238,          357,           121,             1;
  4:         4141,         8282,          4874,           733,            1;
  5:        92292,       230730,        193390,         59355,         4405,            1;
  6:      2512589,      7537767,       8198740,       3834535,       687410,        26437,           1;
  7:     80864308,    283025078,     378474822,     238624360,     69920280,      7768599,      158629,         1;
  8:   3004542341,  12018169364,   19071280147,   15150247667,   6222366045,   1215516903,    86697182,    951781,       1;
  9: 126595099862, 569677949379, 1046242561308, 1003351867476, 532341215532, 151721918691, 20528272772, 961124619, 5710693, 1;

%F Formula

The row polynomials satisfy P(n,x) = (5+5*x)*P(n-2,x) + (5*n-3+(5*n-4)*x)*P(n-1,x) - (5*x+5*x^2)*d/dx 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[(5 + 5*x)*P[n - 2, x] + (5*n - 3 + (5*n - 4)*x)*P[n - 1, x] - (5*x + 5*x^2)*D[P[n - 1, x], 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] = (5 + 5*x)*v[i - 2] + (5*n - 3 + (5*n - 4)*x)*v[i - 1] - (5*x + 5*x^2)*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 - 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

A144442 → x ↦ x/(x+1)

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003