Real-rooted atlas

A038334 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 6\,x + 6\,n\right)\,P_{n}(x) = \left(340 + 528\,x + 144\,x^{2} + 120\,n + 144\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1360 + 2112\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0136021125764048
136800979207603213824224576
292800035328004700160243302433177611925760
322400000111360000211968000188006400729907207962624614687744
452480000032256000008017920000101744640006768230400210213273619110297631004250112
512032000000881664000002709504000004490035200004273274880002274125414405885971660845864714241538338537472
6271360000000231014400000084639744000001734082560000021552168960000164093755392007277201326080161443794124811007531417675349563080704
76041600000000586137600000002494955520000006094061568000009364045824000009310536990720005907375194112002245536409190404358982441369626418075402243652538142556160

All 15 rows

Technical details
Node id
n-155a48d5c10c151b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-98c8469350411ee3

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:             1360,              2112,                576;
  1:            36800,             97920,              76032,              13824;
  2:           928000,           3532800,            4700160,            2433024,              331776;
  3:         22400000,         111360000,          211968000,          188006400,            72990720,             7962624;
  4:        524800000,        3225600000,         8017920000,        10174464000,          6768230400,          2102132736,           191102976;
  5:      12032000000,       88166400000,       270950400000,       449003520000,        427327488000,        227412541440,         58859716608,          4586471424;
  6:     271360000000,     2310144000000,      8463974400000,     17340825600000,      21552168960000,      16409375539200,       7277201326080,       1614437941248,       110075314176;
  7:    6041600000000,    58613760000000,    249495552000000,    609406156800000,     936404582400000,     931053699072000,     590737519411200,     224553640919040,     43589824413696,      2641807540224;
  8:  133120000000000,  1449984000000000,   7033651200000000,  19959644160000000,   36564369408000000,   44947419955200000,   37242147962880000,   20253857808384000,   6736609227571200,   1162395317698560,    63403380965376;
  9: 2908160000000000, 35143680000000000, 191397888000000000, 618961305600000000, 1317336514560000000, 1930598704742400000, 1977686478028800000, 1404561008885760000, 668377307676672000, 197607204008755200, 30687236387241984, 1521681143169024;
%F Formula
The row polynomials satisfy (6*n+11+6*x)*P(n,x) = (120*n+340+(144*n+528)*x+144*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1360 + 2112*x + 576*x^2.
%t Mathematica
P[0, x_] := 1360 + 2112*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((120*n + 340 + (144*n + 528)*x + 144*x^2)*P[n - 1, x])/(6*n + 11 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1360 + 2112*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((120*n + 340 + (144*n + 528)*x + 144*x^2)*v[i - 1])/(6*n + 11 + 6*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. [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.