A253283 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(126 - 253\,n + 167\,n^{2} - 44\,n^{3} + 4\,n^{4}\right)\,P_{n}(x) = \left(-300 + 455\,n - 239\,n^{2} + 52\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(1104 + 4320\,x - 1684\,n - 6648\,n x + 896\,n^{2} + 3568\,n^{2} x - 200\,n^{3} - 800\,n^{3} x + 16\,n^{4} + 64\,n^{4} x\right)\,P_{n-3}(x) + \left(-4320\,x^{2} + 6648\,n x^{2} - 3568\,n^{2} x^{2} + 800\,n^{3} x^{2} - 64\,n^{4} x^{2}\right)\,P_{n-3}'(x) + \left(-1458 - 7776\,x + 34020\,x^{2} + 2268\,n + 12096\,n x - 53568\,n x^{2} - 1242\,n^{2} - 6624\,n^{2} x + 29664\,n^{2} x^{2} + 288\,n^{3} + 1536\,n^{3} x - 6912\,n^{3} x^{2} - 24\,n^{4} - 128\,n^{4} x + 576\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(7776\,x^{2} + 15120\,x^{3} - 12096\,n x^{2} - 23808\,n x^{3} + 6624\,n^{2} x^{2} + 13184\,n^{2} x^{3} - 1536\,n^{3} x^{2} - 3072\,n^{3} x^{3} + 128\,n^{4} x^{2} + 256\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(792 + 3024\,x - 1292\,n - 5064\,n x + 752\,n^{2} + 2992\,n^{2} x - 184\,n^{3} - 736\,n^{3} x + 16\,n^{4} + 64\,n^{4} x\right)\,P_{n-1}(x) + \left(-3024\,x^{2} + 5064\,n x^{2} - 2992\,n^{2} x^{2} + 736\,n^{3} x^{2} - 64\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 4\,x\)

\(P_{3}(x) = 8\,x + 27\,x^{2}\)

\(P_{4}(x) = 12\,x + 108\,x^{2} + 160\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	4														4
3	0	8	27													35
4	0	12	108	160												280
5	0	16	270	960	875											2121
6	0	20	540	3360	7000	4536										15456
7	0	24	945	8960	31500	45360	22638									109427
8	0	28	1512	20160	105000	249480	271656	109824								757660
9	0	32	2268	40320	288750	997920	1765764	1537536	521235							5153825
10	0	36	3240	73920	693000	3243240	8240232	11531520	8339760	2431000						34555948
11	0	40	4455	126720	1501500	9081072	30900870	61501440	70887960	43758000	11177738					228939795
12	0	44	5940	205920	3003000	22702680	98882784	261381120	425327760	415701000	223554760	50791104				1501556112
13	0	48	7722	320320	5630625	51891840	280167888	940972032	2020306860	2771340000	2347324980	1117404288	228501182			9763867785
14	0	52	9828	480480	10010000	110270160	720431712	2979744768	8081227440	14549535000	17213716520	12850149312	5484028368	1019258800		63018862440
15	0	56	12285	698880	17017000	220540320	1711025316	8513556480	28284296040	64017954000	98978869990	102801194496	68550354600	26500728800	4513117500	404109365763

Technical details

Node id: n-7858e7f055f8613b · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e7495ad1473bd81e

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: 0,  4;
   3: 0,  8,   27;
   4: 0, 12,  108,   160;
   5: 0, 16,  270,   960,    875;
   6: 0, 20,  540,  3360,   7000,    4536;
   7: 0, 24,  945,  8960,  31500,   45360,   22638;
   8: 0, 28, 1512, 20160, 105000,  249480,  271656,   109824;
   9: 0, 32, 2268, 40320, 288750,  997920, 1765764,  1537536,  521235;
  10: 0, 36, 3240, 73920, 693000, 3243240, 8240232, 11531520, 8339760, 2431000;

%F Formula

The row polynomials satisfy (4*n^4-44*n^3+167*n^2-253*n+126)*P(n,x) = -(4*n^4-52*n^3+239*n^2-455*n+300)*P(n-4,x) + (16*n^4-200*n^3+896*n^2-1684*n+1104+(64*n^4-800*n^3+3568*n^2-6648*n+4320)*x)*P(n-3,x) - (64*n^4-800*n^3+3568*n^2-6648*n+4320)*x^2*d/dx P(n-3,x) - (24*n^4-288*n^3+1242*n^2-2268*n+1458+(128*n^4-1536*n^3+6624*n^2-12096*n+7776)*x-(576*n^4-6912*n^3+29664*n^2-53568*n+34020)*x^2)*P(n-2,x) + ((128*n^4-1536*n^3+6624*n^2-12096*n+7776)*x^2+(256*n^4-3072*n^3+13184*n^2-23808*n+15120)*x^3)*d/dx P(n-2,x) + (16*n^4-184*n^3+752*n^2-1292*n+792+(64*n^4-736*n^3+2992*n^2-5064*n+3024)*x)*P(n-1,x) - (64*n^4-736*n^3+2992*n^2-5064*n+3024)*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 4*x, P(3,x) = 8*x + 27*x^2, P(4,x) = 12*x + 108*x^2 + 160*x^3.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4*x;
P[3, x_] := 8*x + 27*x^2;
P[4, x_] := 12*x + 108*x^2 + 160*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*P[n - 4, x] + (16*n^4 - 200*n^3 + 896*n^2 - 1684*n + 1104 + (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x)*P[n - 3, x] - (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x^2*D[P[n - 3, x], x] - (24*n^4 - 288*n^3 + 1242*n^2 - 2268*n + 1458 + (128*n^4 - 1536*n^3 + 6624*n^2 - 12096*n + 7776)*x - (576*n^4 - 6912*n^3 + 29664*n^2 - 53568*n + 34020)*x^2)*P[n - 2, x] + ((128*n^4 - 1536*n^3 + 6624*n^2 - 12096*n + 7776)*x^2 + (256*n^4 - 3072*n^3 + 13184*n^2 - 23808*n + 15120)*x^3)*D[P[n - 2, x], x] + (16*n^4 - 184*n^3 + 752*n^2 - 1292*n + 792 + (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x)*P[n - 1, x] - (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x^2*D[P[n - 1, x], x])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; v[3] = 8*x + 27*x^2; v[4] = 12*x + 108*x^2 + 160*x^3; for(n = 5, N, my(i = n); v[i] = (-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*v[i - 4] + (16*n^4 - 200*n^3 + 896*n^2 - 1684*n + 1104 + (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x)*v[i - 3] - (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x^2*deriv(v[i - 3], x) - (24*n^4 - 288*n^3 + 1242*n^2 - 2268*n + 1458 + (128*n^4 - 1536*n^3 + 6624*n^2 - 12096*n + 7776)*x - (576*n^4 - 6912*n^3 + 29664*n^2 - 53568*n + 34020)*x^2)*v[i - 2] + ((128*n^4 - 1536*n^3 + 6624*n^2 - 12096*n + 7776)*x^2 + (256*n^4 - 3072*n^3 + 13184*n^2 - 23808*n + 15120)*x^3)*deriv(v[i - 2], x) + (16*n^4 - 184*n^3 + 752*n^2 - 1292*n + 792 + (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x)*v[i - 1] - (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x^2*deriv(v[i - 1], x))/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)); 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/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

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.