Real-rooted atlas

A085880 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 24\,n x + 9\,n x^{2} + 18\,n^{2} x - 18\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-48 + 108\,x^{2} + 32\,n - 216\,n x - 396\,n x^{2} + 144\,n^{2} x + 108\,n^{2} x^{2} - 162\,n^{2} x^{3} + 72\,n^{3} x^{2} + 108\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
221214
3896162266
440720243021605350
52245376272164838428350109550
61344403202721607257608505003674162257500
78448304128256608091238401603800013856832471517246612500
85491223063042335132810378368024324300031524292821454032660046272962568750
9366080175718402075673601107025920324324000056043187205721075360320246784075996063019863593750
102489344134424576181473177611291664384396972576008574607641611670993734497995515904465095905569569874600409471562500
111719910410319462401567268352011145019392045711993600011848548741122015898917760225686642688016066949464806611913360001200062274848430804687500
12120393728794598604813408851456010727081164805028319296000152056375511043104484333350443444678717440412385036263202545586643600092404795162681499581950048173372773437500
138520171526134523494411387209236481012196376576053377543296000184472789630976439403936412672737891158523904875525153912640720596834496000392364976383072127349113296384186822917943483560986679687500
14608583680047469527040096125792256009398966353920055072068480000021412020225024005828827727923200114197441200128001625975285837760016728140800800000121446302213808005912637403046400173478423804660023214682111320073056669921875000
154381802496036807140966408074566549504086128709861376055512645027840002398146265202688073443229371832320164444315328184320273163848020743680337239318544128000306044681578796160198664616742359040874331255975486402340039956821056028781468127810001497190166015625000
Technical details
Node id
n-484f9a85b0a66d82 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-56805ed74e931bf6

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:       2,        12;
   3:       8,        96,        162;
   4:      40,       720,       2430,        2160;
   5:     224,      5376,      27216,       48384,       28350;
   6:    1344,     40320,     272160,      725760,      850500,      367416;
   7:    8448,    304128,    2566080,     9123840,    16038000,    13856832,      4715172;
   8:   54912,   2306304,   23351328,   103783680,   243243000,   315242928,    214540326,    60046272;
   9:  366080,  17571840,  207567360,  1107025920,  3243240000,  5604318720,   5721075360,  3202467840,   759960630;
  10: 2489344, 134424576, 1814731776, 11291664384, 39697257600, 85746076416, 116709937344, 97995515904, 46509590556, 9569874600;
%F Formula
The row polynomials satisfy (4*n+(18*n^2-24*n)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n,x) = (32*n-48+(144*n^2-216*n)*x+(72*n^3+108*n^2-396*n+108)*x^2+(108*n^3-162*n^2)*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((32*n - 48 + (144*n^2 - 216*n)*x + (72*n^3 + 108*n^2 - 396*n + 108)*x^2 + (108*n^3 - 162*n^2)*x^3)*P[n - 1, x])/(4*n + (18*n^2 - 24*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((32*n - 48 + (144*n^2 - 216*n)*x + (72*n^3 + 108*n^2 - 396*n + 108)*x^2 + (108*n^3 - 162*n^2)*x^3)*v[i - 1])/(4*n + (18*n^2 - 24*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)); 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

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.