Real-rooted atlas

A272867 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 42\,x + 99\,x^{2} + 96\,x^{3} + 33\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16303369
2363601296198010894761
3216324019764626401087029801035937328509
412962592022291210756803183624591624067430884312440118592122667121
5777619440021578401399680058728240166503600323005320423403200359010630177888150391353931564031349
646656139968019035648155131200843482160322276320088711718401772519760025515335340258099534001741880764870443707401291467969107918163081
7279936979776015774393615480460801034300534449760373600177744838320478837526400977596610760150525130140017208175141081416558916080793900582398271208273490426184429777446353252589

All 15 rows

Technical details
Node id
n-1df5db9823996fe6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9217fe7884f21aef

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:        6,        30,         33;
  2:       36,       360,       1296,         1980,          1089;
  3:      216,      3240,      19764,        62640,        108702,         98010,          35937;
  4:     1296,     25920,     222912,      1075680,       3183624,       5916240,        6743088,         4312440,         1185921;
  5:     7776,    194400,    2157840,     13996800,      58728240,     166503600,      323005320,       423403200,       359010630,        177888150,         39135393;
  6:    46656,   1399680,   19035648,    155131200,     843482160,    3222763200,     8871171840,     17725197600,     25515335340,      25809953400,      17418807648,       7044370740,       1291467969;
  7:   279936,   9797760,  157743936,   1548046080,   10343005344,   49760373600,   177744838320,    478837526400,    977596610760,    1505251301400,    1720817514108,    1416558916080,     793900582398,     271208273490,      42618442977;
  8:  1679616,  67184640, 1249634304,  14343920640,  113704964352,  659937922560,  2900599414272,   9847462636800,  26096285121120,   54161044502400,   87743132281728,  109797171865920,  104047148942352,   72190711343520,   34590678081696,   10228426314480,    1406408618241;
  9: 10077696, 453496320, 9568772352, 125769646080, 1153785337344, 7844125847040, 40953997986048, 167880709693440, 547721370501696, 1432821087662400, 3012467537759328, 5078391468226560, 6813721414928736, 7177865407907040, 5806821323893392, 3481384374673920, 1456783617839814, 379730326925070, 46411484401953;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+42*x+99*x^2+96*x^3+33*x^4)*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[((6 + 42*x + 99*x^2 + 96*x^3 + 33*x^4)*P[n - 1, x])/(1 + 2*x + 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] = ((6 + 42*x + 99*x^2 + 96*x^3 + 33*x^4)*v[i - 1])/(1 + 2*x + 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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.