Real-rooted atlas

A038243 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + n x\right)\,P_{n}(x) = \left(18 + 12\,x + 2\,x^{2} + 6\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
236481296
321643221632896
4129634562592768808192
57776259202592011520240019273728
646656186624233280138240432006912448655360
72799361306368195955214515206048001451521881610245767168
81679616895795215676416139345927257600232243245158449152230450331648
910077696604661761209323521254113287838208031352832812851213271041244165120436207616
10604661764031078409069926401074954240783820800376233984121927680265420803732480307200112643758096384
1136279705626605117446651279360886837248073903104004138573824160944537643794432082114560101376007434242457632212254720
1221767823361741425868847889211392709469798406651279360042568187904193133445126306398208147806208024330240026763264176947253248274877906944
1313060694016113192681472339578044416553386442752576444211200415039832064215205838848819831767042305776844847443968006958448646900940841533441146882336462209024
14783641640967313988648962377046310912422586010828848421313740803873705099264225966130790498379812044832280875827279705866240146127421441932263424174440448963379224576019791209299968
Technical details
Node id
n-e02fd4d802592fb7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-20d8219123bfe5ea

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:        6,        4;
  2:       36,       48,        12;
  3:      216,      432,       216,        32;
  4:     1296,     3456,      2592,       768,       80;
  5:     7776,    25920,     25920,     11520,     2400,      192;
  6:    46656,   186624,    233280,    138240,    43200,     6912,     448;
  7:   279936,  1306368,   1959552,   1451520,   604800,   145152,   18816,    1024;
  8:  1679616,  8957952,  15676416,  13934592,  7257600,  2322432,  451584,   49152,   2304;
  9: 10077696, 60466176, 120932352, 125411328, 78382080, 31352832, 8128512, 1327104, 124416, 5120;
%F Formula
The row polynomials satisfy (3+n*x)*P(n,x) = (18+(6*n+12)*x+(2*n+2)*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[((18 + (6*n + 12)*x + (2*n + 2)*x^2)*P[n - 1, x])/(3 + n*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] = ((18 + (6*n + 12)*x + (2*n + 2)*x^2)*v[i - 1])/(3 + n*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

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.