Real-rooted atlas

A130123 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,x + 6\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 12\,x + 24\,x^{2} + 16\,x^{3} + 12\,n x + 40\,n x^{2} + 32\,n x^{3} + 8\,n^{2} x^{2} + 16\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
121618
2464144212
3819286410242088
41651234568192640018576
532128011520409606400036864154656
6643072345601638403840004423682007041228608
712871689676857344017920003096576280985610485769424512
82561638425804818350087168000165150722247884816777216530841670357248
951236864663552550502425804800743178241348730881509949449555148826214400513962496
101024819201658880157286408601600029727129667436544010066329609555148805242880001268776963688436736
11204818022440550404325376027033600010899947522967207936553648128070071091205767168000279130931260397977626081077248
124096393216973209611534336081100800037371248641186883174426575110144420426547204613734400033495711744144955146242835349504182124122112
138192851968230031362998927362342912000121456558084408423219211515881062421862180454429989273600029029616844818844169011273719087104131533373441258180190208
141638418350085367398476336332865601536003778648473615429481267246063524249610202350878721679399321600203207317913617587891077121032067219456368293445632603979776008611350921216
Technical details
Node id
n-06a6c84bb2b2c046 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-181bf1738e48841f

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:   2,    16;
  2:   4,    64,    144;
  3:   8,   192,    864,    1024;
  4:  16,   512,   3456,    8192,     6400;
  5:  32,  1280,  11520,   40960,    64000,    36864;
  6:  64,  3072,  34560,  163840,   384000,   442368,    200704;
  7: 128,  7168,  96768,  573440,  1792000,  3096576,   2809856,   1048576;
  8: 256, 16384, 258048, 1835008,  7168000, 16515072,  22478848,  16777216,  5308416;
  9: 512, 36864, 663552, 5505024, 25804800, 74317824, 134873088, 150994944, 95551488, 26214400;
%F Formula
The row polynomials satisfy (1+(6*n-2)*x+4*n^2*x^2)*P(n,x) = (2+(12*n+12)*x+(8*n^2+40*n+24)*x^2+(16*n^2+32*n+16)*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[((2 + (12*n + 12)*x + (8*n^2 + 40*n + 24)*x^2 + (16*n^2 + 32*n + 16)*x^3)*P[n - 1, x])/(1 + (6*n - 2)*x + 4*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] = ((2 + (12*n + 12)*x + (8*n^2 + 40*n + 24)*x^2 + (16*n^2 + 32*n + 16)*x^3)*v[i - 1])/(1 + (6*n - 2)*x + 4*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.