Real-rooted atlas

A038324 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(21 + 20\,n x\right)\,P_{n}(x) = \left(441 + 840\,x + 400\,x^{2} + 420\,n x + 400\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1214061
2441168012003321
39261529207560032000169781
41944811481760317520026880008000008339441
54084101388962001111320001411200008400000019200000398432301
685766121980184240350065800059270400005292000000241920000044800000018652848361
71801088541240145138801029193452002178187200002593080000001778112000006585600000010240000000859768867621
837822859361576348333120288174166560073187089920001089093600000099574272000005531904000000172032000000023040000000039145609050081
979428004658113616229369960778070249712002305393332480004116773808000004704884352000003485099520000001625702400000004354560000000051200000000001764668475635741
101667988097820131771201863240020424344054940006916179997440000144087083280000001976051427840000018296772480000000113799168000000004572288000000000107520000000000011264000000000078899046188944601
113502775005422217339147630408440524224830743460001997046974260800004754873748240000007607797997184000008453108885760000006571901952000000003520661760000000001241856000000000002601984000000000024576000000000003503314079949776661
12735582751138664116813320026026608013210465734735192005591731527930240000149778523069560000002738807278986240000035503057320192000000331223858380800000002218016908800000000010431590400000000000327849984000000000061931520000000000053248000000000000154642457912265811921
13154472377739119461382503030592105332032785974050751885600152654270712495552000454328186644332000000934617983954054400000138461923548748800000015070685556326400000001211037232204800000000711956044800000000000298343485440000000000845365248000000000001453670400000000000011468800000000000006791610580410222010381
143243919932521508681865045315339068981608032563642434211972004080032326315790208000133572486873433608000003053085414249911040000050884756904165184000000632968793365708800000005934082437803520000000041863015434240000000000219282461798400000000008284579430400000000000213689548800000000000033718272000000000000024576000000000000000296958095843119375012041
Technical details
Node id
n-4659453931b6128d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9974e62f5594415f

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:           21,             40;
  2:          441,           1680,           1200;
  3:         9261,          52920,          75600,           32000;
  4:       194481,        1481760,        3175200,         2688000,          800000;
  5:      4084101,       38896200,      111132000,       141120000,        84000000,        19200000;
  6:     85766121,      980184240,     3500658000,      5927040000,      5292000000,      2419200000,       448000000;
  7:   1801088541,    24014513880,   102919345200,    217818720000,    259308000000,    177811200000,     65856000000,     10240000000;
  8:  37822859361,   576348333120,  2881741665600,   7318708992000,  10890936000000,   9957427200000,   5531904000000,   1720320000000,   230400000000;
  9: 794280046581, 13616229369960, 77807024971200, 230539333248000, 411677380800000, 470488435200000, 348509952000000, 162570240000000, 43545600000000, 5120000000000;
%F Formula
The row polynomials satisfy (21+20*n*x)*P(n,x) = (441+(420*n+840)*x+(400*n+400)*x^2)*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[((441 + (420*n + 840)*x + (400*n + 400)*x^2)*P[n - 1, x])/(21 + 20*n*x)]];
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] = ((441 + (420*n + 840)*x + (400*n + 400)*x^2)*v[i - 1])/(21 + 20*n*x)); 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.