A257624 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 12\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(-100 + 272\,x - 156\,x^{2} - 210\,n - 396\,n x - 108\,n x^{2} - 90\,n^{2} - 1080\,n^{2} x + 1053\,n^{2} x^{2} - 540\,n^{3} x + 1782\,n^{3} x^{2} + 729\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-444\,x + 1084\,x^{2} - 740\,x^{3} - 630\,n x + 2277\,n x^{2} - 2532\,n x^{3} - 270\,n^{2} x + 1944\,n^{2} x^{2} - 2457\,n^{2} x^{3} + 540\,n^{3} x^{2} - 270\,n^{3} x^{3} + 324\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-144\,x^{2} + 348\,x^{3} - 204\,x^{4} - 189\,n x^{2} + 945\,n x^{3} - 756\,n x^{4} - 81\,n^{2} x^{2} + 972\,n^{2} x^{3} - 891\,n^{2} x^{4} + 324\,n^{3} x^{3} - 324\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(40 + 32\,x + 102\,n + 144\,n x + 63\,n^{2} + 216\,n^{2} x + 108\,n^{3} x\right)\,P_{n-1}(x) + \left(60\,x - 80\,x^{2} + 99\,n x - 288\,n x^{2} + 54\,n^{2} x - 324\,n^{2} x^{2} - 108\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 5 + 20\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	20														25
2	25	320	225													570
3	125	3660	8235	2000												14020
4	625	36280	181170	145120	15625											378820
5	3125	332740	3135690	5574560	2079625	112500										11238240
6	15625	2911920	47219895	156087680	131166375	26207280	765625									364374400
7	78125	24732860	650455245	3612317680	5644246375	2601820980	302977535	5000000								12841628800
8	390625	205987880	8434933200	73699745120	191904376750	165824426520	45923525200	3295806080	31640625							489320832000
9	1953125	1693215540	104834556180	1376691478080	5565491439750	8014307673240	4216117651620	745490177280	34287614685	195312500						20059111072000
10	9765625	13795724320	1263662431965	24119366845440	144088204469250	320572306929600	282412880759730	96477467381760	11372961887685	344893108000	1181640625					880666730944000
11	48828125	111732982060	14893578902655	402819954550800	3429348754073250	11183647470613560	15222186835001790	8779132810427520	2039276019913425	165484210029500	3379922707315	7031250000				41240288369280000
12	244140625	901285731480	172628340266430	6486758519062240	76551708009828375	352067766764773680	700220594410082340	625898252026264320	248027533951843935	40542240744139000	2320892130248670	32446286333280	41259765625			2052321763972480000
13	1220703125	7250324914340	1975971672972270	101555938216782560	1625716962120194375	10238154661785906540	28564360750829717940	37308552817410243840	23035847989018289715	6502867848480777500	768016782764418110	31615546767556320	306326227630865	239257812500		108178979086097920000
14	6103515625	58217443064720	22404530876042835	1555270999164638080	33191028727273600625	279676130850216687600	1061103013224577212435	1940044746505332679680	1754068246350831718515	776878141250601910000	160644579040004227025	13997438992481742720	420707302005693235	2852654710171280	1373291015625	6021604620024913920000

Technical details

Node id: n-d5f69f3763822c3a · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-fee45ca65514c000

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:       5,         20;
  2:      25,        320,          225;
  3:     125,       3660,         8235,          2000;
  4:     625,      36280,       181170,        145120,         15625;
  5:    3125,     332740,      3135690,       5574560,       2079625,        112500;
  6:   15625,    2911920,     47219895,     156087680,     131166375,      26207280,        765625;
  7:   78125,   24732860,    650455245,    3612317680,    5644246375,    2601820980,     302977535,      5000000;
  8:  390625,  205987880,   8434933200,   73699745120,  191904376750,  165824426520,   45923525200,   3295806080,    31640625;
  9: 1953125, 1693215540, 104834556180, 1376691478080, 5565491439750, 8014307673240, 4216117651620, 745490177280, 34287614685, 195312500;

%F Formula

The row polynomials satisfy (9*n^2+12*n+4)*P(n,x) = -(90*n^2+210*n+100+(540*n^3+1080*n^2+396*n-272)*x-(729*n^4+1782*n^3+1053*n^2-108*n-156)*x^2)*P(n-2,x) - ((270*n^2+630*n+444)*x-(540*n^3+1944*n^2+2277*n+1084)*x^2-(324*n^4-270*n^3-2457*n^2-2532*n-740)*x^3)*d/dx P(n-2,x) - ((81*n^2+189*n+144)*x^2-(324*n^3+972*n^2+945*n+348)*x^3+(324*n^3+891*n^2+756*n+204)*x^4)*d^2/dx^2 P(n-2,x) + (63*n^2+102*n+40+(108*n^3+216*n^2+144*n+32)*x)*P(n-1,x) + ((54*n^2+99*n+60)*x-(108*n^3+324*n^2+288*n+80)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 20*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 20*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(90*n^2 + 210*n + 100 + (540*n^3 + 1080*n^2 + 396*n - 272)*x - (729*n^4 + 1782*n^3 + 1053*n^2 - 108*n - 156)*x^2)*P[n - 2, x] - ((270*n^2 + 630*n + 444)*x - (540*n^3 + 1944*n^2 + 2277*n + 1084)*x^2 - (324*n^4 - 270*n^3 - 2457*n^2 - 2532*n - 740)*x^3)*D[P[n - 2, x], x] - ((81*n^2 + 189*n + 144)*x^2 - (324*n^3 + 972*n^2 + 945*n + 348)*x^3 + (324*n^3 + 891*n^2 + 756*n + 204)*x^4)*D[P[n - 2, x], {x, 2}] + (63*n^2 + 102*n + 40 + (108*n^3 + 216*n^2 + 144*n + 32)*x)*P[n - 1, x] + ((54*n^2 + 99*n + 60)*x - (108*n^3 + 324*n^2 + 288*n + 80)*x^2)*D[P[n - 1, x], x])/(9*n^2 + 12*n + 4)]];
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] = 5 + 20*x; for(n = 2, N, my(i = n + 1); v[i] = (-(90*n^2 + 210*n + 100 + (540*n^3 + 1080*n^2 + 396*n - 272)*x - (729*n^4 + 1782*n^3 + 1053*n^2 - 108*n - 156)*x^2)*v[i - 2] - ((270*n^2 + 630*n + 444)*x - (540*n^3 + 1944*n^2 + 2277*n + 1084)*x^2 - (324*n^4 - 270*n^3 - 2457*n^2 - 2532*n - 740)*x^3)*deriv(v[i - 2], x) - ((81*n^2 + 189*n + 144)*x^2 - (324*n^3 + 972*n^2 + 945*n + 348)*x^3 + (324*n^3 + 891*n^2 + 756*n + 204)*x^4)*deriv(deriv(v[i - 2], x), x) + (63*n^2 + 102*n + 40 + (108*n^3 + 216*n^2 + 144*n + 32)*x)*v[i - 1] + ((54*n^2 + 99*n + 60)*x - (108*n^3 + 324*n^2 + 288*n + 80)*x^2)*deriv(v[i - 1], x))/(9*n^2 + 12*n + 4)); 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 5.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 5.?
    
    ↑ Liu–Wang theorem (above)

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.