Real-rooted atlas

A130123 → basis A013611 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(10 + 32\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1103242
210064010241764
310009600307203276874088
410000128000614400131072010485763111696
5100000160000010240000327680005242880033554432130691232
61000000192000001536000006553600001572864000201326592010737418245489031744
71000000022400000021504000001146880000036700160000704643072007516192768034359738368230539333248
810000000025600000002867200000018350080000073400320000018790481920003006477107200274877906944010995116277769682651996416
910000000002880000000036864000000027525120000001321205760000042278584320000901943132160001236950581248009895604649984035184372088832406671383849472
10100000000003200000000004608000000000393216000000002202009600000008455716864000002254857830400000412316860416000049478023249920003518437208883200112589990684262417080198121677824
111000000000003520000000000563200000000005406720000000003460300800000000155021475840000004960687226880000011338713661440000018141941858304000019351404648857600012384898975268864036028797018963968717368321110468608
1210000000000003840000000000067584000000000072089600000000005190451200000000026575110144000000099213744537600000027212912787456000005442582557491200000774056185954304000074309393851613184004323455642275676160115292150460684697630129469486639681536
131000000000000041600000000000079872000000000009371648000000000074973184000000000043184553984000000001842540969984000000058961311039488000000141507146494771200000251568260435148800000322007373356990464000281024616747918950400149879795598890106880368934881474191032321265437718438866624512
14100000000000000448000000000000093184000000000000119275520000000000010496245760000000000671759728640000000003224446697472000000001179226220789760000000330183341821132800000070439112921841664000001127025806749466624000013114482114902884352000104915856919223074816005165088340638674452480118059162071741130342453148384174432398229504
Technical details
Node id
n-839abbd637a9ac81 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-833b492313a93e6e

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:         10,          32;
  2:        100,         640,         1024;
  3:       1000,        9600,        30720,         32768;
  4:      10000,      128000,       614400,       1310720,        1048576;
  5:     100000,     1600000,     10240000,      32768000,       52428800,       33554432;
  6:    1000000,    19200000,    153600000,     655360000,     1572864000,     2013265920,     1073741824;
  7:   10000000,   224000000,   2150400000,   11468800000,    36700160000,    70464307200,    75161927680,     34359738368;
  8:  100000000,  2560000000,  28672000000,  183500800000,   734003200000,  1879048192000,  3006477107200,   2748779069440,  1099511627776;
  9: 1000000000, 28800000000, 368640000000, 2752512000000, 13212057600000, 42278584320000, 90194313216000, 123695058124800, 98956046499840, 35184372088832;
%F Formula
The row polynomials satisfy P(n,x) = (10+32*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[(10 + 32*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] = (10 + 32*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.