Real-rooted atlas

A038259 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 20\,n x\right)\,P_{n}(x) = \left(121 + 440\,x + 400\,x^{2} + 220\,n x + 400\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1114051
212188012002201
3133114520396003200087451
41464121296087120014080008000003306801
5161051292820015972000387200004400000019200000120981251
6177156138652240263538000851840000145200000012672000004480000004323001801
71948717149603708040584852001639792000037268000000487872000003449600000010240000000151763129451
8214358881623589472059524449600288603392000819896000000143109120000015178240000009011200000002304000000005254909295201
9235794769177169197160841845787200476195596800016233940800000354195072000005008819200000044605440000000228096000000005120000000000179960008900051
102593742460194317907640011575379574000748307366400002976222480000007792291584000001377425280000000163553280000000012545280000000005632000000000001126400000000006107552719115001
1128531167061111412466824440155624547606000113181489168000051446131440000001571445469440000033333691776000000494748672000000005059929600000000034073600000000000136294400000000002457600000000000205726700032181051
12313842837672113694960189328020542440283992001659995174464000084886116876000000296329717094400000733341219072000000130613649408000000016697767680000000001499238400000000000899543040000000000324403200000000000532480000000000006885697238925709201
13345227121439311631982755894920267051723691896002373793099483520001348746079252000000529689369306240000014981113475328000000311295864422400000004775561556480000000053597772800000000000428782182400000000002319482880000000000076144640000000000001146880000000000000229209870082467980451
1437974983358324119332718800601360342716378737933200332331033927692800020770689620480800000906357365257344000002883864344000640000006848509017292800000001225727466163200000000165081140224000000000016508114022400000000001190667878400000000000586313728000000000000176619520000000000000245760000000000000007593856898505408245801
Technical details
Node id
n-753fa8e2d561ab51 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-54ba3237eacbac96

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:         11,          40;
  2:        121,         880,         1200;
  3:       1331,       14520,        39600,         32000;
  4:      14641,      212960,       871200,       1408000,         800000;
  5:     161051,     2928200,     15972000,      38720000,       44000000,       19200000;
  6:    1771561,    38652240,    263538000,     851840000,     1452000000,     1267200000,      448000000;
  7:   19487171,   496037080,   4058485200,   16397920000,    37268000000,    48787200000,    34496000000,    10240000000;
  8:  214358881,  6235894720,  59524449600,  288603392000,   819896000000,  1431091200000,  1517824000000,   901120000000,   230400000000;
  9: 2357947691, 77169197160, 841845787200, 4761955968000, 16233940800000, 35419507200000, 50088192000000, 44605440000000, 22809600000000, 5120000000000;
%F Formula
The row polynomials satisfy (11+20*n*x)*P(n,x) = (121+(220*n+440)*x+(400*n+400)*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[((121 + (220*n + 440)*x + (400*n + 400)*x^2)*P[n - 1, x])/(11 + 20*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] = ((121 + (220*n + 440)*x + (400*n + 400)*x^2)*v[i - 1])/(11 + 20*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

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.