Real-rooted atlas

A038291 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(10\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2601878
3900540811521
41200010800324032426364
515000018000081000162001215428415
61800000270000016200004860007290043746683274
721000000378000002835000011340000255150030618015309101362989
824000000050400000045360000022680000068040000122472001224720524881505964408
92700000000648000000068040000004082400000153090000036741600055112400472392017714722024729467
1030000000000810000000009720000000068040000000306180000009185400000183708000023619600017714700590490318134981190
11330000000000990000000000133650000000010692000000005613300000002020788000005051970000086605200009743085006495390019486174549330231017
123600000000000118800000000001782000000000016038000000000962280000000040415760000001212472800000259815600000389723400003897234000233834040637729264517774185332
13390000000000001404000000000002316600000000002316600000000001563705000000007505784000000026270244000000675520560000012666010500001688801400001519921260082904796020726199908625319776759
14420000000000000163800000000000029484000000000003243240000000000243243000000000013135122000000005254048800000001576214640000003546482940000059108049000007092965880005803335720029016678606696156612720754476874626
Technical details
Node id
n-6a5f903d0d9009ce · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-1024c0779b1dc97d

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:          0;
  1:          3;
  2:         60,         18;
  3:        900,        540,         81;
  4:      12000,      10800,       3240,        324;
  5:     150000,     180000,      81000,      16200,       1215;
  6:    1800000,    2700000,    1620000,     486000,      72900,      4374;
  7:   21000000,   37800000,   28350000,   11340000,    2551500,    306180,    15309;
  8:  240000000,  504000000,  453600000,  226800000,   68040000,  12247200,  1224720,   52488;
  9: 2700000000, 6480000000, 6804000000, 4082400000, 1530900000, 367416000, 55112400, 4723920, 177147;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (10*n+3*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((10*n + 3*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((10*n + 3*n*x)*v[i - 1])/(n - 1)); 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.