Real-rooted atlas

A038284 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 + 60\,x + 36\,x^{2} + 36\,n x + 36\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(128 + 864\,x + 1224\,x^{2} + 486\,x^{3} + 288\,n x + 648\,n x^{2} + 324\,n x^{3} + 72\,n^{2} x^{2} + 54\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 64 + 384\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
064384324772
151246087776345616352
240964915212441611059232400320656
3327684915201658880221184012960002799365970944
42621444718592199065603538944031104000134369282286144107103808
5209715244040192222953472495452160580608000376233984128024064179159041867324928
61677721640265318423781703686341787648928972800080263249924096770048114661785613604889631834878208
713421772836238786562446118092876101451776133772083200144473849856983224811524127824281697955205121007769600532970676224
810737418243221225472024461180928086973087744017836277760002311581597696196644962304011007531417603918208204808062156800073164072968789799617536
98589934592283467841536239175991296095670396518402242274918400033903196766208346095133655042421656911872011493410734080354734899200064384384204852242776064143139732119552
10687194767362473901162496229608951644161020484229529602690729902080004649581270794245537522138480644649581270794242758418576179201135151677440003090450441830450153065021443678762147842305938109468672
11549755813888214404767416322170848270090241061303598710784310928788684800060444556520325128227175748599808805927420271001657375106384527362951394361344000107135615316787226079593811148838259126337536255989602713636802448261906432
124398046511104184717953466368202612505208422410806000277782528348240243326976007521989255862681611518046048039731212894838724336025610710019858445107266111233694105600299979722887004169736381689495552214251107490201628670835503923217631936921600582586643568541696
13351843720888321583296743997440187026927884697601080600027778252803798984472657920009026387107035217921535739473071964160193422580865040384018360034043048755201322224673882112000719951334928809984292091450684866560857004429960806401720250130235392021158324305920001203673560514569156293624259805184
1428147497671065613510798882111488170996048351723520106397541196627968040522501041684480001050343227000461721619657465255321141248275089892785835212802937605446887800832024177822608130048000153589618118146129927477541137532583936274241417587458048073397338890043392013541327555788800015407021574586368815300763254784142989303831280549888
Technical details
Node id
n-30e58548258a254c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e997b32cda4979c2

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:         64,          384,           324;
  1:        512,         4608,          7776,          3456;
  2:       4096,        49152,        124416,        110592,          32400;
  3:      32768,       491520,       1658880,       2211840,        1296000,         279936;
  4:     262144,      4718592,      19906560,      35389440,       31104000,       13436928,        2286144;
  5:    2097152,     44040192,     222953472,     495452160,      580608000,      376233984,      128024064,       17915904;
  6:   16777216,    402653184,    2378170368,    6341787648,     9289728000,     8026324992,     4096770048,     1146617856,      136048896;
  7:  134217728,   3623878656,   24461180928,   76101451776,   133772083200,   144473849856,    98322481152,    41278242816,     9795520512,    1007769600;
  8: 1073741824,  32212254720,  244611809280,  869730877440,  1783627776000,  2311581597696,  1966449623040,  1100753141760,   391820820480,   80621568000,   7316407296;
  9: 8589934592, 283467841536, 2391759912960, 9567039651840, 22422749184000, 33903196766208, 34609513365504, 24216569118720, 11493410734080, 3547348992000, 643843842048, 52242776064;
%F Formula
The row polynomials satisfy (16+(36*n+60)*x+(9*n^2+36*n+36)*x^2)*P(n,x) = (128+(288*n+864)*x+(72*n^2+648*n+1224)*x^2+(54*n^2+324*n+486)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 384*x + 324*x^2.
%t Mathematica
P[0, x_] := 64 + 384*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((128 + (288*n + 864)*x + (72*n^2 + 648*n + 1224)*x^2 + (54*n^2 + 324*n + 486)*x^3)*P[n - 1, x])/(16 + (36*n + 60)*x + (9*n^2 + 36*n + 36)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 64 + 384*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((128 + (288*n + 864)*x + (72*n^2 + 648*n + 1224)*x^2 + (54*n^2 + 324*n + 486)*x^3)*v[i - 1])/(16 + (36*n + 60)*x + (9*n^2 + 36*n + 36)*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.