Real-rooted atlas

A027467 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(256 - 16\,x + 48\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(4096 + 768\,x + 48\,x^{2} + x^{3} + 768\,n x + 80\,n x^{2} + 2\,n x^{3} + 16\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
116420
22561289393
340963072432167616
4655366553613824102425145945
510485761310720368640409602000362770932
61677721625165824884736013107209600034564952200625
7268435456469762048198180864367001603584000193536548864976861616
8429496729685899345924227858432939524096114688000825753635123281928118175589457
968719476736154618822656869730877442254857830433030144002972712961685913658982411664100336477711860
1010995116277762748779069440173946175488051539607552088080384000951268147267436544031457280933120160001216201448365049
1117592186044416483785116221443401614098432011338713661440221459251200027903865651223737663488138412032054743040140800021296144113844361437120
12281474976710656844424930131968653109906898944241892558110720531502202880007653631721472759605231616531502202882627665920901120002044416276481692082521699163817
134503599627370496146366987889541121234971460318003250313652087029761228360646656000198994424758272225711268823041842540969984109310902272468582400014174617628753923515219637973261108857364
14720575940379279362522015791327477762305280059260272641024568915226787842751527848509440049527501273169926319915527045125896131103948840809403514882099249152007937785856214695936393702443904225690407351117266785
Technical details
Node id
n-236d62986d3f18f5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8ecf423fa3129810

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:          16,            4;
  2:         256,          128,           9;
  3:        4096,         3072,         432,          16;
  4:       65536,        65536,       13824,        1024,         25;
  5:     1048576,      1310720,      368640,       40960,       2000,        36;
  6:    16777216,     25165824,     8847360,     1310720,      96000,      3456,       49;
  7:   268435456,    469762048,   198180864,    36700160,    3584000,    193536,     5488,     64;
  8:  4294967296,   8589934592,  4227858432,   939524096,  114688000,   8257536,   351232,   8192,    81;
  9: 68719476736, 154618822656, 86973087744, 22548578304, 3303014400, 297271296, 16859136, 589824, 11664, 100;
%F Formula
The row polynomials satisfy (256+(48*n-16)*x+n^2*x^2)*P(n,x) = (4096+(768*n+768)*x+(16*n^2+80*n+48)*x^2+(n^2+2*n+1)*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[((4096 + (768*n + 768)*x + (16*n^2 + 80*n + 48)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(256 + (48*n - 16)*x + 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] = ((4096 + (768*n + 768)*x + (16*n^2 + 80*n + 48)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(256 + (48*n - 16)*x + 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

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.