Real-rooted atlas

A038327 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(169 - 52\,x + 156\,n x + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2197 + 2028\,x + 624\,x^{2} + 64\,x^{3} + 2028\,n x + 1040\,n x^{2} + 128\,n x^{3} + 208\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1131629
2169416144729
3219781125616102416949
428561140608146016532486400374833
5371293228488031636801730560416000368648003277
64826809356441286169176044994560162240002875392200704166457353
762748517540602608112279003210236262404921280001308303361826406410485763392038373

All 15 rows

Technical details
Node id
n-3269ecd1e431c86f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-83244bdbb25f2a59

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:          13,           16;
  2:         169,          416,          144;
  3:        2197,         8112,         5616,         1024;
  4:       28561,       140608,       146016,        53248,         6400;
  5:      371293,      2284880,      3163680,      1730560,       416000,        36864;
  6:     4826809,     35644128,     61691760,     44994560,     16224000,      2875392,      200704;
  7:    62748517,    540602608,   1122790032,   1023626240,    492128000,    130830336,    18264064,    1048576;
  8:   815730721,   8031810176,  19461693888,  21291425792,  12795328000,   4535451648,   949731328,  109051904,   5308416;
  9: 10604499373, 117465223824, 325288312128, 415182802944, 299410675200, 132661960704, 37039521792, 6379536384, 621084672, 26214400;
%F Formula
The row polynomials satisfy (169+(156*n-52)*x+16*n^2*x^2)*P(n,x) = (2197+(2028*n+2028)*x+(208*n^2+1040*n+624)*x^2+(64*n^2+128*n+64)*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[((2197 + (2028*n + 2028)*x + (208*n^2 + 1040*n + 624)*x^2 + (64*n^2 + 128*n + 64)*x^3)*P[n - 1, x])/(169 + (156*n - 52)*x + 16*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] = ((2197 + (2028*n + 2028)*x + (208*n^2 + 1040*n + 624)*x^2 + (64*n^2 + 128*n + 64)*x^3)*v[i - 1])/(169 + (156*n - 52)*x + 16*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.