Real-rooted atlas

A038271 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(144 + 120\,x + 25\,x^{2} + 60\,n + 25\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
117522
226417025459
33888396012751259248
45529677760396008500625181781
576723213824009720003300005312531253507882
6104509442301696020736000972000024750003187501562566733279
714034124836578304040279680024192000085050000173250001859375781251255153588
8186325401656136499207315660800537062400024192000006804000001155000001062500039062523389304361
9245089566728384643072012630712320010973491200060419520000217728000005103000000742500000597656251953125432496961342
10319906381824122544783360020961607680002105118720000137168640000060419520000018144000000036450000000464062500033203125097656257945387725299
11414846340300817594851000320336998154240003842961408000028945382400000150885504000005538456000000142560000000025059375000028359375000182617187548828125145151560832328
12534966026895362489078041804805278455300096006739963084800005764442112000003473445888000001508855040000004747248000000010692000000000167062500000017015625000099609375002441406252638936015687741
136865397345157123477279174819840808950363586560011436653150208000109524400128000007493774745600000376289971200000014010796800000003857139000000007722000000000010859062500000100546875000053955078125122070312547775023452840402
1487734428410839044805778141609984012170477111869440018875508483686400020014143012864000015333416017920000087427372032000000376289971200000001225944720000000029999970000000005405400000000006910312500000058652343750002905273437506103515625861698288862816519
Technical details
Node id
n-9619a6c9dae2f70e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-29c6f18c0c8a36b8

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:          17,           5;
  2:         264,         170,           25;
  3:        3888,        3960,         1275,          125;
  4:       55296,       77760,        39600,         8500,         625;
  5:      767232,     1382400,       972000,       330000,       53125,        3125;
  6:    10450944,    23016960,     20736000,      9720000,     2475000,      318750,      15625;
  7:   140341248,   365783040,    402796800,    241920000,    85050000,    17325000,    1859375,     78125;
  8:  1863254016,  5613649920,   7315660800,   5370624000,  2419200000,   680400000,  115500000,  10625000,   390625;
  9: 24508956672, 83846430720, 126307123200, 109734912000, 60419520000, 21772800000, 5103000000, 742500000, 59765625, 1953125;
%F Formula
The row polynomials satisfy (5*n+7+5*x)*P(n,x) = (60*n+144+(25*n+120)*x+25*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[((60*n + 144 + (25*n + 120)*x + 25*x^2)*P[n - 1, x])/(5*n + 7 + 5*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] = ((60*n + 144 + (25*n + 120)*x + 25*x^2)*v[i - 1])/(5*n + 7 + 5*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

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.