Real-rooted atlas

A038291 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(10\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
28032112
312009601922352
416000192007680102443904
5200000320000192000512005120768320
624000004800000384000015360003072002457612907776
728000000672000006720000035840000107520001720320114688210827008
832000000089600000010752000007168000002867200006881280091750405242883373232128
936000000001152000000016128000000129024000006451200000206438400041287680047185920235929653128406016
10400000000001440000000002304000000002150400000001290240000005160960000013762560000235929600023592960010485760826441871360
1144000000000017600000000003168000000000337920000000023654400000001135411200000378470400000865075200001297612800011534336004613734412727204818944
124800000000000211200000000004224000000000050688000000000405504000000002270822400000090832896000002595225600000519045120000692060160005536481280201326592194379128143872
135200000000000024960000000000054912000000000073216000000000065894400000000042172416000000019680460800000067475865600000168689664000002998927360000359871283200261724569608724152322948083443515392
14560000000000000291200000000000069888000000000001025024000000000010250240000000000738017280000000039360921600000001574436864000000472331059200000104962457600000167939932160001832071987200122138132480375809638444448027302232064
Technical details
Node id
n-29c1a5ee5bc02216 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9629d1e9c04dbfbe

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:          4;
  2:         80,          32;
  3:       1200,         960,         192;
  4:      16000,       19200,        7680,        1024;
  5:     200000,      320000,      192000,       51200,       5120;
  6:    2400000,     4800000,     3840000,     1536000,     307200,      24576;
  7:   28000000,    67200000,    67200000,    35840000,   10752000,    1720320,    114688;
  8:  320000000,   896000000,  1075200000,   716800000,  286720000,   68812800,   9175040,   524288;
  9: 3600000000, 11520000000, 16128000000, 12902400000, 6451200000, 2064384000, 412876800, 47185920, 2359296;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (10*n+4*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((10*n + 4*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((10*n + 4*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.