Real-rooted atlas

A038221 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,x + 6\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(6 + 36\,x + 72\,x^{2} + 48\,x^{3} + 36\,n x + 120\,n x^{2} + 96\,n x^{3} + 24\,n^{2} x^{2} + 48\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
164854
23657612961908
32165184233282764856376
41296414722799366635525184001504656
577763110402799360995328015552000895795237581408
646656223948825194240119439360279936000322486272146313216895655232
7279936156764162116316161254113280391910400067722117126145155072229323571220611407744
816796161074954241693052928120394874884702924800010835538739214748372172811007531417634828517376461613904128
91007769672559411213060694016108355387392507915878400146279772979226547069911042972033482752188073993830451597803520010116323808768
1060466176483729408097955205120928760463360507915878400017553572757504398206048665605944066965504056422198149120309586821120007492001071104217798500824064
1136279705631926140928718338170880766227382272047889211392000193089300332544525631984238592980771049308160124128835928064010216365096960004944720706928641069932053790724620184591251456
12217678233620897110425651720348303366129819058176043100290252800019860613748490246307583810863104141231031100375042234319046705152024519276232704000178009945449431047703510787293184150682097575526496788225579323392
131306069401613583121776643667442879692847812588653772837353584885760001936409840477798470284505321046016183600340430487552348553771286003712478125886537728000462825858168520704300436920704434176117532036108910592209706682542981122005945615392989184
14783641640968776786378752256721001578496365114313356083231377011304038400180731585111261184737987305870983168220320408516585062448797527980040519688032514893833830400971934302153893478484122337797241569284936345516574244864176153613336104140828888165452349440041187824504297570304
Technical details
Node id
n-94c7e7b4a817d6c6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-181bf1738e48841f

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:        6,        48;
  2:       36,       576,        1296;
  3:      216,      5184,       23328,        27648;
  4:     1296,     41472,      279936,       663552,       518400;
  5:     7776,    311040,     2799360,      9953280,     15552000,       8957952;
  6:    46656,   2239488,    25194240,    119439360,    279936000,     322486272,     146313216;
  7:   279936,  15676416,   211631616,   1254113280,   3919104000,    6772211712,    6145155072,    2293235712;
  8:  1679616, 107495424,  1693052928,  12039487488,  47029248000,  108355387392,  147483721728,  110075314176,   34828517376;
  9: 10077696, 725594112, 13060694016, 108355387392, 507915878400, 1462797729792, 2654706991104, 2972033482752, 1880739938304, 515978035200;
%F Formula
The row polynomials satisfy (1+(6*n-2)*x+4*n^2*x^2)*P(n,x) = (6+(36*n+36)*x+(24*n^2+120*n+72)*x^2+(48*n^2+96*n+48)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((6 + (36*n + 36)*x + (24*n^2 + 120*n + 72)*x^2 + (48*n^2 + 96*n + 48)*x^3)*P[n - 1, x])/(1 + (6*n - 2)*x + 4*n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((6 + (36*n + 36)*x + (24*n^2 + 120*n + 72)*x^2 + (48*n^2 + 96*n + 48)*x^3)*v[i - 1])/(1 + (6*n - 2)*x + 4*n^2*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.