Real-rooted atlas

A147716 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(15 + 4\,n x\right)\,P_{n}(x) = \left(225 + 120\,x + 16\,x^{2} + 60\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115823
222524048513
333755400216025611191
45062510800064800153601280240065
5759375202500016200005760009600061445082519
611390625364500003645000017280000432000055296028672106472257
71708593756378750007654500004536000001512000002903040030105601310722211156407
82562890625109350000001530900000010886400000453600000011612160001806336001572864058982445587458689
9384433593751845281250002952450000002449440000001224720000003919104000081285120001061683200796262402621440934095967255
10576650390625307546875000055358437500005248800000000306180000000011757312000003048192000005308416000059719680003932160001153433619038574168961
11864975585937550745234375000101490468750000108256500000000721710000000003233260800000010059033600000218972160000032845824000032440320000190316544050331648386257174241463
121297463378906258303765625000001826828437500000216513000000000016238475000000008314099200000003017710080000007882997760000014780620800000194641920000017128488960090596966402181038087804847346180673
131946195068359375134936191406250003238468593750000042220035000000000351833625000000002026561680000000084064780800000002561974272000000576444211200000948879360000001113351782400088332042240042530242560939524096157145359253697431
14291929260253906252179738476562500005667320039062500008060188500000000007388506125000000004728643920000000002206700496000000007685922816000000020175547392000000398529331200000058450968576000061832429568000446567546880019730006016040265318403153973759669279425
Technical details
Node id
n-8786392edd324db1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-cba14508bff4210d

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:          15,            8;
  2:         225,          240,           48;
  3:        3375,         5400,         2160,          256;
  4:       50625,       108000,        64800,        15360,         1280;
  5:      759375,      2025000,      1620000,       576000,        96000,        6144;
  6:    11390625,     36450000,     36450000,     17280000,      4320000,      552960,      28672;
  7:   170859375,    637875000,    765450000,    453600000,    151200000,    29030400,    3010560,     131072;
  8:  2562890625,  10935000000,  15309000000,  10886400000,   4536000000,  1161216000,  180633600,   15728640,   589824;
  9: 38443359375, 184528125000, 295245000000, 244944000000, 122472000000, 39191040000, 8128512000, 1061683200, 79626240, 2621440;
%F Formula
The row polynomials satisfy (15+4*n*x)*P(n,x) = (225+(60*n+120)*x+(16*n+16)*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[((225 + (60*n + 120)*x + (16*n + 16)*x^2)*P[n - 1, x])/(15 + 4*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] = ((225 + (60*n + 120)*x + (16*n + 16)*x^2)*v[i - 1])/(15 + 4*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.