Real-rooted atlas

A027466 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + n x\right)\,P_{n}(x) = \left(16 + 16\,x + 4\,x^{2} + 8\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18816
26412848240
3512153611522563456
4409616384184328192128048384
532768163840245760163840512006144663552
62621441572864294912026214401228800294912286728957952
720971521468006433030144367001602293760082575361605632131072119439360
8167772161342177283523215364697620483670016001761607685138022483886085898241576599552
913421772812079595523623878656563714457652848230403170893824123312537630198988842467328262144020639121408
1010737418241073741824036238786560644245094407046430720050734301184246625075208053063680169869312020971520011534336268308578304
118589934592944892805123543348019207086696038408858370048007441030840324340601323521771674009604982833152092274688001015021568503316483467372396544
126871947673682463372083234016140984327559142440960106300440576001020484229529669449621176323401614098432119587995648029527900160048721035264483183820821810380844580502241280
135497558138887146825580544321607151124487861508138598412283606466560013266294983884810318229431910458961311039488248743030947847677254041600168899588915225125558681622682796032939524096570630428688384
14439804651110461572651155456300166674382848800444465020928137576392425472016509167091056641444552120467456943380976631808464320324435968171970490531840472918848962569380208574464127023657779210522669875240265318407275537965776896
Technical details
Node id
n-b1b7a75e45782fe8 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e620d4a356abb37c

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:         8,          8;
  2:        64,        128,         48;
  3:       512,       1536,       1152,        256;
  4:      4096,      16384,      18432,       8192,       1280;
  5:     32768,     163840,     245760,     163840,      51200,       6144;
  6:    262144,    1572864,    2949120,    2621440,    1228800,     294912,      28672;
  7:   2097152,   14680064,   33030144,   36700160,   22937600,    8257536,    1605632,    131072;
  8:  16777216,  134217728,  352321536,  469762048,  367001600,  176160768,   51380224,   8388608,   589824;
  9: 134217728, 1207959552, 3623878656, 5637144576, 5284823040, 3170893824, 1233125376, 301989888, 42467328, 2621440;
%F Formula
The row polynomials satisfy (2+n*x)*P(n,x) = (16+(8*n+16)*x+(4*n+4)*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[((16 + (8*n + 16)*x + (4*n + 4)*x^2)*P[n - 1, x])/(2 + 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] = ((16 + (8*n + 16)*x + (4*n + 4)*x^2)*v[i - 1])/(2 + 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.