Real-rooted atlas

A038324 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 - 110\,x + 330\,n x + 100\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 3630\,x + 3300\,x^{2} + 1000\,x^{3} + 3630\,n x + 5500\,n x^{2} + 2000\,n x^{3} + 1100\,n^{2} x^{2} + 1000\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1114051
21218809001901
3133114520297001600061551
4146412129606534007040002500001835001
51610512928200119790001936000013750000360000051778251
6177156138652240197653500425920000453750000237600000490000001404347301
71948717149603708030438639008198960000116462500009147600000377300000064000000036965198151
8214358881623589472044643337200144301696000256217500000268329600000166012000000563200000008100000000950374386801
92357947691771691971606313843404002380977984000507310650000066411576000005478396000000278784000000080190000000010000000000023974289569251
10259374246019431790764008681534680500374153683200009300695250000014610546720000015065589000000010222080000000044104500000000110000000000001210000000000595369629201501
1128531167061111412466824440116718410704500565907445840000160769160750000029464602552000003645872538000000309217920000000017788815000000006655000000000001464100000000001440000000000014591718735739551
1231384283767211369496018932801540683021299400829997587232000026526911523750000555618219552000008020919583600000081633530880000000587030895000000002928200000000000096630600000000001900800000000000169000000000000353630156618839401
133452271214393116319827558949202002887927689220011868965497417600042148314976625000099316756744920000016385592863640000001945599152640000000167890835970000000010468315000000000004606058600000000001359072000000000002416700000000000019600000000000008487574115638557051
1437974983358324119332718800601360257037284053449900166165516963846400064908405064002500001699420059857520000031542266262507000000428031813580800000004309198123230000000032242410200000000000177333256100000000006976569600000000000186085900000000000030184000000000000022500000000000000201998379290188548501
Technical details
Node id
n-53030eeff73af9a5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4b3bb4c5c4994d07

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:         11,          40;
  2:        121,         880,          900;
  3:       1331,       14520,        29700,         16000;
  4:      14641,      212960,       653400,        704000,        250000;
  5:     161051,     2928200,     11979000,      19360000,      13750000,       3600000;
  6:    1771561,    38652240,    197653500,     425920000,     453750000,     237600000,      49000000;
  7:   19487171,   496037080,   3043863900,    8198960000,   11646250000,    9147600000,    3773000000,     640000000;
  8:  214358881,  6235894720,  44643337200,  144301696000,  256217500000,  268329600000,  166012000000,   56320000000,   8100000000;
  9: 2357947691, 77169197160, 631384340400, 2380977984000, 5073106500000, 6641157600000, 5478396000000, 2787840000000, 801900000000, 100000000000;
%F Formula
The row polynomials satisfy (121+(330*n-110)*x+100*n^2*x^2)*P(n,x) = (1331+(3630*n+3630)*x+(1100*n^2+5500*n+3300)*x^2+(1000*n^2+2000*n+1000)*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[((1331 + (3630*n + 3630)*x + (1100*n^2 + 5500*n + 3300)*x^2 + (1000*n^2 + 2000*n + 1000)*x^3)*P[n - 1, x])/(121 + (330*n - 110)*x + 100*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] = ((1331 + (3630*n + 3630)*x + (1100*n^2 + 5500*n + 3300)*x^2 + (1000*n^2 + 2000*n + 1000)*x^3)*v[i - 1])/(121 + (330*n - 110)*x + 100*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.