Real-rooted atlas

A038208 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 6\,x + 18\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(16 + 72\,x + 108\,x^{2} + 54\,x^{3} + 72\,n x + 180\,n x^{2} + 108\,n x^{3} + 36\,n^{2} x^{2} + 54\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
142428
216192324532
3641152388834568560
42566144311045529632400125200
51024307202073605529606480002799361720000
640961474561244160442368077760006718464228614422600000
71638468812869672963096576072576000940584966401203217915904287200000
865536314572837158912198180864580608000100329062410241925125733089281360488963556000000
92621441415577619110297611890851844180377600902961561612290310144103195607044897760256100776960043120000000
101048576629145609555148806794772480278691840007223692492812290310144013759414272097955205120403107840007316407296514000000000
114194304276824064467140608037371248640175177728000529737449472108154729267215135355699201436676341760886837248000321921921024522427760646040000000000
12167772161207959552224227491841993133260801051066368000363248536780886523783413761452994147123217240116101120141893959680007726126104576250765325107236787621478470120000000000
1367108864523449139210599845068810364292956166072827904000236111548907526427481053593612592615941734417929720745164818446214758400013391951914598465198984527872191295631687682559896027136805600000000000
142684354562254857830449465943654452763673231363400783626240014691385265356844992367375155210074092753387521673440602882048206597605294080018748732680437761217047711186944535627768725504143354177519616176319369216009172000000000000
Technical details
Node id
n-f963c29f77333339 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-56805ed74e931bf6

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      4,       24;
  2:     16,      192,       324;
  3:     64,     1152,      3888,       3456;
  4:    256,     6144,     31104,      55296,      32400;
  5:   1024,    30720,    207360,     552960,     648000,     279936;
  6:   4096,   147456,   1244160,    4423680,    7776000,    6718464,     2286144;
  7:  16384,   688128,   6967296,   30965760,   72576000,   94058496,    64012032,    17915904;
  8:  65536,  3145728,  37158912,  198180864,  580608000, 1003290624,  1024192512,   573308928,  136048896;
  9: 262144, 14155776, 191102976, 1189085184, 4180377600, 9029615616, 12290310144, 10319560704, 4897760256, 1007769600;
%F Formula
The row polynomials satisfy (4+(18*n-6)*x+9*n^2*x^2)*P(n,x) = (16+(72*n+72)*x+(36*n^2+180*n+108)*x^2+(54*n^2+108*n+54)*x^3)*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 + (72*n + 72)*x + (36*n^2 + 180*n + 108)*x^2 + (54*n^2 + 108*n + 54)*x^3)*P[n - 1, x])/(4 + (18*n - 6)*x + 9*n^2*x^2)]];
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 + (72*n + 72)*x + (36*n^2 + 180*n + 108)*x^2 + (54*n^2 + 108*n + 54)*x^3)*v[i - 1])/(4 + (18*n - 6)*x + 9*n^2*x^2)); 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.