Real-rooted atlas

A027467 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(15 + 31\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1151631
2225480256961
333751080011520409629791
45062521600034560024576065536923521
57593754050000864000092160004915200104857628629151
611390625729000001944000002764800002211840009437184016777216887503681
717085937512757500004082400000725760000077414400004954521600176160768026843545627512614111
825628906252187000000081648000000174182400000232243200000198180864000105696460800322122547204294967296852891037441
9384433593753690562500001574640000000391910400000062705664000006688604160000475634073600021743271936005798205849606871947673626439622160671
105766503906256150937500000295245000000008398080000000015676416000000020065812480000017836277760000010871635968000043486543872000103079215104001099511627776819628286980801
1186497558593751014904687500005412825000000001732104000000000369515520000000055180984320000005885971660800000448454983680000023917599129600008504035246080001814194185830401759218604441625408476896404831
1212974633789062516607531250000009743085000000000346420800000000008314099200000000014189395968000000017657914982400000016144379412480000010762919608320000051024211476480000163277476724736003166593487994880281474976710656787662783788549761
13194619506835937526987238281250000172718325000000000675520560000000000180138816000000000034586652672000000004918990602240000000524692330905600000041975386472448000002487430309478400000106130359871078400030874286507950080054887620458577920450359962737049624417546297445042591
142919292602539062543594769531250000030225706875000000001289630160000000000037829151360000000000807021895680000000001291235033088000000001574076992716800000001469138526535680000001044720729980928000005571843893231616000021612000555565056000576320014815068160094575592174780416072057594037927936756943935220796320321
Technical details
Node id
n-748ef5bd7f0b1a24 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d935ec63feee39a0

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:          15,           16;
  2:         225,          480,           256;
  3:        3375,        10800,         11520,          4096;
  4:       50625,       216000,        345600,        245760,         65536;
  5:      759375,      4050000,       8640000,       9216000,       4915200,       1048576;
  6:    11390625,     72900000,     194400000,     276480000,     221184000,      94371840,      16777216;
  7:   170859375,   1275750000,    4082400000,    7257600000,    7741440000,    4954521600,    1761607680,     268435456;
  8:  2562890625,  21870000000,   81648000000,  174182400000,  232243200000,  198180864000,  105696460800,   32212254720,   4294967296;
  9: 38443359375, 369056250000, 1574640000000, 3919104000000, 6270566400000, 6688604160000, 4756340736000, 2174327193600, 579820584960, 68719476736;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (15+31*x+16*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[((15 + 31*x + 16*x^2)*P[n - 1, x])/(1 + 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] = ((15 + 31*x + 16*x^2)*v[i - 1])/(1 + 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.