A144442 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(5\,x + 15\,x^{2} + 15\,x^{3} + 5\,x^{4}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(5\,x + 15\,x^{2} + 15\,x^{3} + 5\,x^{4}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

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

Technical details

Node id: n-e0355357b9c46242 · revision 0 · JSON
Definition source: derived:mobius_x_over_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: 1,       2;
  2: 1,      19,        19;
  3: 1,     121,       357,         238;
  4: 1,     733,      4874,        8282,         4141;
  5: 1,    4405,     59355,      193390,       230730,        92292;
  6: 1,   26437,    687410,     3834535,      8198740,      7537767,       2512589;
  7: 1,  158629,   7768599,    69920280,    238624360,    378474822,     283025078,      80864308;
  8: 1,  951781,  86697182,  1215516903,   6222366045,  15150247667,   19071280147,   12018169364,   3004542341;
  9: 1, 5710693, 961124619, 20528272772, 151721918691, 532341215532, 1003351867476, 1046242561308, 569677949379, 126595099862;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5*x+15*x^2+15*x^3+5*x^4)*P(n-2,x) + (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (5*x+15*x^2+15*x^3+5*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*x + 15*x^2 + 15*x^3 + 5*x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (5*x + 15*x^2 + 15*x^3 + 5*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*x + 15*x^2 + 15*x^3 + 5*x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (5*x + 15*x^2 + 15*x^3 + 5*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^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, 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) = (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+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