Real-rooted atlas

A038319 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(32 + 30\,x + 16\,n + 15\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 480 + 450\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0480450930
111520216001012543245
22457606912006480002025001787460
34915200184320002592000016200000379687569264075
494371840442368000829440000777600000364500000683437502576623590
5176160768099090432002322432000029030400000204120000007654500000119601562593187886505
632212254720211392921600594542592000928972800000870912000000489888000000153090000000205031250003301513693320
7579820584960434865438720014269022208000267544166400003135283200000023514624000000110224800000002952450000000345990234375115140290054535

All 15 rows

Technical details
Node id
n-98095ae5bcb61b1c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d935ec63feee39a0

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:             480,              450;
  1:           11520,            21600,            10125;
  2:          245760,           691200,           648000,            202500;
  3:         4915200,         18432000,         25920000,          16200000,           3796875;
  4:        94371840,        442368000,        829440000,         777600000,         364500000,          68343750;
  5:      1761607680,       9909043200,      23224320000,       29030400000,       20412000000,        7654500000,        1196015625;
  6:     32212254720,     211392921600,     594542592000,      928972800000,      870912000000,      489888000000,      153090000000,       20503125000;
  7:    579820584960,    4348654387200,   14269022208000,    26754416640000,    31352832000000,    23514624000000,    11022480000000,     2952450000000,     345990234375;
  8:  10307921510400,   86973087744000,  326149079040000,   713451110400000,  1003290624000000,   940584960000000,   587865600000000,   236196000000000,   55358437500000,    5766503906250;
  9: 181419418583040, 1700807049216000, 7175279738880000, 17938199347200000, 29429858304000000, 33108590592000000, 25866086400000000, 13856832000000000, 4871542500000000, 1014904687500000, 95147314453125;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (16*n+32+(15*n+30)*x)*P(n-1,x) for n >= 1, with P(0,x) = 480 + 450*x.
%t Mathematica
P[0, x_] := 480 + 450*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 32 + (15*n + 30)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 480 + 450*x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 32 + (15*n + 30)*x)*v[i - 1])/(n + 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.