Real-rooted atlas

A038334 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(9 + 30\,x + 16\,x^{2} + 18\,n x + 16\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(108 + 648\,x + 816\,x^{2} + 288\,x^{3} + 216\,n x + 432\,n x^{2} + 192\,n x^{3} + 48\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 768\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01447685761488
117281382420736819244480
2207362211844976643932161024001235200
324883233177609953280117964806144000117964832640000
42985984477757441791590402831155202211840008493465612845056832000000
5358318086688604163009871872594542592061931520003567255552107898470413421772820633600000
642998169691729428484815794995211415217766414863564800011415217766451791265792128849018881358954496500736000000
7515978035212383472844874300837068820547391979523210529996800308210879692818644855685126957847019521467670855681342177280011939840000000
861917364224165112971264011145125560320352241005363206421059993600073970611126272559345670553602783138807808088060251340801610612736000129922760704280576000000000
9743008370688217949122068481634618415513605811976588492801210828455936000162735344477798414766725702615049184358065766403874651058995201063004405760001714980441292812369505812486512640000000000
1089161004482562853152143441922353850518339584929916254158848021794912206848000334769851497185283544014168627609626450951229407232139487438123827205102421147648000123478591773081617812088369971211613591568384149585920000000000
11106993205379072370909778647449633381880078270464145066935648780288377778478252032000652801210419511296789808871865581568687724731964588032435200806946340864198994424758272000642088677220024321389342892857753618117202846679041077521395220483404595200000000000
121283918464548864479329560098242564673463210957864962215567744454098944634667843463413760012185622594497544192165859863091772129281650539356715011276812185622594497544192668621267187793920026967724443241021447780320200003420161521845039121039361810235943970406498956046499840076873728000000000000
131540702157458636861628086298345472064709490613262745603323351616681148416010385473802128588800021934120670095579545633171972618354425856037137135526087753728031334458100136542208020058638015633817600097083807995667677184350114409000153907209131070234726236160162921234957336576017812088369971200090071992547409921723596800000000000000
14184884258895036416788839504618822041688744444269617479680490833469540600381440166167580834057420800038285010624166829752326369018742724049764352792258891223205412864075202699440327701299205501797855716704256000310668185586136566988813444393305605910036484382913712668593356801042695903726954086401709960483517235200017293822569102704648134626826937958438404096000000000000000
Technical details
Node id
n-6e5997629441d544 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e543af4f498eab8b

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:          144,            768,             576;
  1:         1728,          13824,           20736,            8192;
  2:        20736,         221184,          497664,          393216,           102400;
  3:       248832,        3317760,         9953280,        11796480,          6144000,          1179648;
  4:      2985984,       47775744,       179159040,       283115520,        221184000,         84934656,         12845056;
  5:     35831808,      668860416,      3009871872,      5945425920,       6193152000,       3567255552,       1078984704,       134217728;
  6:    429981696,     9172942848,     48157949952,    114152177664,     148635648000,     114152177664,      51791265792,     12884901888,      1358954496;
  7:   5159780352,   123834728448,    743008370688,   2054739197952,    3210529996800,    3082108796928,    1864485568512,    695784701952,    146767085568,     13421772800;
  8:  61917364224,  1651129712640,  11145125560320,  35224100536320,   64210599936000,   73970611126272,   55934567055360,  27831388078080,   8806025134080,   1610612736000,   129922760704;
  9: 743008370688, 21794912206848, 163461841551360, 581197658849280, 1210828455936000, 1627353444777984, 1476672570261504, 918435806576640, 387465105899520, 106300440576000, 17149804412928, 1236950581248;
%F Formula
The row polynomials satisfy (9+(18*n+30)*x+(4*n^2+16*n+16)*x^2)*P(n,x) = (108+(216*n+648)*x+(48*n^2+432*n+816)*x^2+(32*n^2+192*n+288)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 768*x + 576*x^2.
%t Mathematica
P[0, x_] := 144 + 768*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((108 + (216*n + 648)*x + (48*n^2 + 432*n + 816)*x^2 + (32*n^2 + 192*n + 288)*x^3)*P[n - 1, x])/(9 + (18*n + 30)*x + (4*n^2 + 16*n + 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 768*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((108 + (216*n + 648)*x + (48*n^2 + 432*n + 816)*x^2 + (32*n^2 + 192*n + 288)*x^3)*v[i - 1])/(9 + (18*n + 30)*x + (4*n^2 + 16*n + 16)*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

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.