Real-rooted atlas

A013611 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-11 + 16\,x + 16\,n\right)\,P_{n}(x) = \left(25 + 160\,x + 256\,x^{2} + 80\,n + 256\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1211637
21856722561113
31325888016128409630429
486258480028416034406465536787185
553125690000339200075776006881280104857619642581
631562551000003312000010854400018186240013212057616777216477839817
7182812535350000285600000123648000030392320004073717760246625075226843545611406894093
810390625234000000226240000012185600000395673600007780433920086905978880450971566084294967296268362192609
958203125149625000016848000000108595200000438681600000113953996800018673041408001787780136960811748818944687194767366240771794565
103222656259312500000119700000000898560000000434380800000014037811200000303877324800004268123750400035755602739200144310901145601099511627776143764688431161
11176757812556718750000819500000000702240000000039536640000000152902041600000411775795200000764034416640000938987225088000699220675788800253987186016256175921860444163285936552705597
1296191406253393750000005445000000000524480000000003370752000000001518206976000000489286533120000011294421811200000183368259993600002003172746854400013425036975144960443323088319283228147497671065674609107615493073
1352001953125200078125000035295000000000377520000000000272729600000000014022328320000000526311751680000001453879984128000002936549670912000004237844230963200004166599313457152002538552446209228807684266864200908845035996273704961684484500107540789
142795410156251164843750000022408750000000026353600000000002114112000000000012218286080000000052350025728000000016841976053760000004070863955558400000730874584760320000094927710773575680008484711329221836800473863123292389376013240582904469258247205759403792793637845732546084267945
Technical details
Node id
n-b87f0ed547d456fa · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-003c76dbd54fc355

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:       21,         16;
  2:      185,        672,         256;
  3:     1325,       8880,       16128,         4096;
  4:     8625,      84800,      284160,       344064,        65536;
  5:    53125,     690000,     3392000,      7577600,      6881280,       1048576;
  6:   315625,    5100000,    33120000,    108544000,    181862400,     132120576,      16777216;
  7:  1828125,   35350000,   285600000,   1236480000,   3039232000,    4073717760,    2466250752,     268435456;
  8: 10390625,  234000000,  2262400000,  12185600000,  39567360000,   77804339200,   86905978880,   45097156608,   4294967296;
  9: 58203125, 1496250000, 16848000000, 108595200000, 438681600000, 1139539968000, 1867304140800, 1787780136960, 811748818944, 68719476736;
%F Formula
The row polynomials satisfy (16*n-11+16*x)*P(n,x) = (80*n+25+(256*n+160)*x+256*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[((80*n + 25 + (256*n + 160)*x + 256*x^2)*P[n - 1, x])/(16*n - 11 + 16*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] = ((80*n + 25 + (256*n + 160)*x + 256*x^2)*v[i - 1])/(16*n - 11 + 16*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.