Real-rooted atlas

A038338 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(24 + 84\,x + 60\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 2880\,x + 3600\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0576288036007056
113824103680259200216000592704
2331776331776012441600207360001296000049787136
3796262499532800497664000124416000015552000007776000004182119424
41911029762866544640179159040005971968000011197440000011197440000046656000000351298031616
54586471424802632499206019743744002508226560000627056640000094058496000007838208000000279936000000029509034655744
6110075314176220150628352019263179980800963158999040003009871872000006019743744000007524679680000005374771200000001679616000000002478758911082496
72641807540224594406696550405944066965504003467372396544000130026464870400003250661621760000054177693696000000580475289600000003627970560000000010077696000000000208215748530929664

All 15 rows

Technical details
Node id
n-66894599312bd669 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9629d1e9c04dbfbe

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:              576,              2880,               3600;
  1:            13824,            103680,             259200,              216000;
  2:           331776,           3317760,           12441600,            20736000,             12960000;
  3:          7962624,          99532800,          497664000,          1244160000,           1555200000,            777600000;
  4:        191102976,        2866544640,        17915904000,         59719680000,         111974400000,         111974400000,           46656000000;
  5:       4586471424,       80263249920,       601974374400,       2508226560000,        6270566400000,        9405849600000,         7838208000000,         2799360000000;
  6:     110075314176,     2201506283520,     19263179980800,      96315899904000,      300987187200000,      601974374400000,       752467968000000,       537477120000000,       167961600000000;
  7:    2641807540224,    59440669655040,    594406696550400,    3467372396544000,    13002646487040000,    32506616217600000,     54177693696000000,     58047528960000000,     36279705600000000,     10077696000000000;
  8:   63403380965376,  1585084524134400,  17832200896512000,  118881339310080000,   520105859481600000,  1560317578444800000,   3250661621760000000,   4643802316800000000,   4353564672000000000,   2418647040000000000,    604661760000000000;
  9: 1521681143169024, 41846231437148160, 523077892964352000, 3923084197232640000, 19615420986163200000, 68653973451571200000, 171634933628928000000, 306490952908800000000, 383113691136000000000, 319261409280000000000, 159630704640000000000, 36279705600000000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (24+84*x+60*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 2880*x + 3600*x^2.
%t Mathematica
P[0, x_] := 576 + 2880*x + 3600*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24 + 84*x + 60*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] = 576 + 2880*x + 3600*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24 + 84*x + 60*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. [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.