Real-rooted atlas

A038323 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(100 - 90\,x + 270\,n x + 81\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2000 + 5400\,x + 4860\,x^{2} + 1458\,x^{3} + 5400\,n x + 8100\,n x^{2} + 2916\,n x^{3} + 1620\,n^{2} x^{2} + 1458\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1207292
2400288029166196
380008640017496093312362672
4160000230400069984007464960262440019551760
532000005760000023328000037324800026244000068024448997792448
6640000001382400000699840000014929920000157464000008162933760166659897648950652736
7128000000032256000000195955200000522547200000734832000000571405363200233323856640391820820482330781701888
8256000000007372800000005225472000000167215104000002939328000000030474952704000186659085312006269133127680892616806656108405753569536
951200000000016588800000000134369280000000501645312000000105815808000000013713728716800001119954511872000564221981491200160671025198080198359290368004947329791278080
101024000000000036864000000000033592320000000001433272320000000035271936000000000548549148672000005599772559360000037614798766080000160671025198080003967185807360000432026534421504222276525288469504
11204800000000000811008000000000082114560000000000394149888000000000110854656000000000020113468784640000002463899926118400000206881393213440000011782541847859200004363904388096000009504583757273088092546510514094089856131736936460288
124096000000000000176947200000000000197074944000000000010510663680000000000332563968000000000006896046440448000000098555997044736000000993030687424512000007069525108715520000034911235104768000000114055005087277056002221116252338257920195504503461023744432166990768117387264
13819200000000000003833856000000000000465813504000000000002732772556800000000009607403520000000000002241215093145600000000366065131880448000000043031329788395520000003676153056532070400000226923028180992000000098847671075640115200028874511280397352960050831170899866173440408130111367154892818767031758785534803968
1416384000000000000008257536000000000000010868981760000000000006956148326400000000000269007298560000000000006972669178675200000000012812279615815680000000017212531915358208000000017155380930482995200000012707689578135552000000069193369752948080640000269495438617041960960007116363925981264281600114276431182803369984084333007705968230400808119877163243894947840
Technical details
Node id
n-8b25bad2cda1991a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-048d14b6dbcd016e

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:           20,             72;
  2:          400,           2880,            2916;
  3:         8000,          86400,          174960,           93312;
  4:       160000,        2304000,         6998400,         7464960,          2624400;
  5:      3200000,       57600000,       233280000,       373248000,        262440000,         68024448;
  6:     64000000,     1382400000,      6998400000,     14929920000,      15746400000,       8162933760,       1666598976;
  7:   1280000000,    32256000000,    195955200000,    522547200000,     734832000000,     571405363200,     233323856640,     39182082048;
  8:  25600000000,   737280000000,   5225472000000,  16721510400000,   29393280000000,   30474952704000,   18665908531200,   6269133127680,    892616806656;
  9: 512000000000, 16588800000000, 134369280000000, 501645312000000, 1058158080000000, 1371372871680000, 1119954511872000, 564221981491200, 160671025198080, 19835929036800;
%F Formula
The row polynomials satisfy (100+(270*n-90)*x+81*n^2*x^2)*P(n,x) = (2000+(5400*n+5400)*x+(1620*n^2+8100*n+4860)*x^2+(1458*n^2+2916*n+1458)*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[((2000 + (5400*n + 5400)*x + (1620*n^2 + 8100*n + 4860)*x^2 + (1458*n^2 + 2916*n + 1458)*x^3)*P[n - 1, x])/(100 + (270*n - 90)*x + 81*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] = ((2000 + (5400*n + 5400)*x + (1620*n^2 + 8100*n + 4860)*x^2 + (1458*n^2 + 2916*n + 1458)*x^3)*v[i - 1])/(100 + (270*n - 90)*x + 81*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

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.