Real-rooted atlas

A038291 → basis A013609 → basis A013611

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(12 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112820
214419264400
31728345623045128000
4207365529655296245764096160000
52488328294401105920737280245760327683200000
629859841194393619906560176947208847360235929626214464000000
735831808167215104334430208371589120247726080990904322202009620971521280000000
84299816962293235712535088332871345111045945425920317089382410569646082013265921677721625600000000
9515978035230958682112825564856321284211998721284211998728561413324838050725888108716359681811939328134217728512000000000
10619173642244127824281601238347284480220150628352025684239974402054739197952114152177664043486543872010871635968016106127360107374182410240000000000
117430083706885448728051712181624268390403632485367808048433138237440452042623549443013617490329614350559477760478351982592010630044057601417339207688589934592204800000000000
1289161004482567132880358604826153894648217658119765884928087179648827392092991625415884872326819767910441329611295948817220671373312051024211476480102048422952961236950581248687194767364096000000000000
13106993205379072927274446618624370909778647449690666834780487681511113913008128018133366956097536161185484054200321074569893694668853728494684733441989944247582720530651799355392964821453373441072023837081654975581388881920000000000000
1412839184645488641198323900245606451927369010642944138472984028381184253867137385365504338489516513820672338489516513820672257896774486720512150439785117253632668621267187793922228737557292646454030001388912649005000231485449235897673318443980465111041638400000000000000
Technical details
Node id
n-ac10b3f602dabf0c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f06e51c139cfae05

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:         12,           8;
  2:        144,         192,          64;
  3:       1728,        3456,        2304,          512;
  4:      20736,       55296,       55296,        24576,         4096;
  5:     248832,      829440,     1105920,       737280,       245760,       32768;
  6:    2985984,    11943936,    19906560,     17694720,      8847360,     2359296,      262144;
  7:   35831808,   167215104,   334430208,    371589120,    247726080,    99090432,    22020096,     2097152;
  8:  429981696,  2293235712,  5350883328,   7134511104,   5945425920,  3170893824,  1056964608,   201326592,   16777216;
  9: 5159780352, 30958682112, 82556485632, 128421199872, 128421199872, 85614133248, 38050725888, 10871635968, 1811939328, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (12+8*x)*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[(12 + 8*x)*P[n - 1, 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] = (12 + 8*x)*v[i - 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.