Real-rooted atlas

A136572 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
151217
250240288578
37505400129601036829478
4150001440005184008294404976642004504
5375000450000021600000518400006220800029859840170382840
611250000162000000972000000311040000055987200005374771200214990848017379049680
73937500006615000000476280000001905120000004572288000006584094720005267275776001805923123202068106911920

All 15 rows

Technical details
Node id
n-32b7f037deab8913 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f055f948fcf55643

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:            5,             12;
  2:           50,            240,             288;
  3:          750,           5400,           12960,           10368;
  4:        15000,         144000,          518400,          829440,           497664;
  5:       375000,        4500000,        21600000,        51840000,         62208000,         29859840;
  6:     11250000,      162000000,       972000000,      3110400000,       5598720000,       5374771200,        2149908480;
  7:    393750000,     6615000000,     47628000000,    190512000000,     457228800000,     658409472000,      526727577600,      180592312320;
  8:  15750000000,   302400000000,   2540160000000,  12192768000000,   36578304000000,   70230343680000,    84276412416000,    57789539942400,   17336861982720;
  9: 708750000000, 15309000000000, 146966400000000, 823011840000000, 2962842624000000, 7110822297600000, 11377315676160000, 11702381838336000, 7021429103001600, 1872381094133760;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+12*n*x)*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[(5*n + 12*n*x)*P[n - 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] = (5*n + 12*n*x)*v[i - 1]); 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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.