Real-rooted atlas

A178820 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(15 + 27\,x + 5\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1203656
22509008101960
3250013500243001458054880
4218751575004252505103002296351344560
517500015750005670000102060009185400330674430118144
61312500141750006378750015309000020667150014880348044641044632481024
79375000118125000637875000191362500034445250003720087000223205220057395628012649620480
864453125928125000584718750021049875000473622187506820159500061381435500315675954007102708965243505194240
942968750069609375005011875000021049875000056834662500010230239250001227628710000947027862000426162537900852325075804545430292480
102792968750502734375004072148437501954631250000615708843750013299311025000199489665375002051893701000013850282481750554011299270099722033868682726831323136
11177734375003519140625003167226562500171030234375006157088437500015515862862500027928553152500035908139767500032317325790750019390395474450069805423708020114227056976761474041721757696
1211108398437523994140625002375419921875014252519531250057722704101562516624138781250003491069144062500538622096512500060594985857656254847598868612500261770338905075085670292732570012850543909885525795730130759680
13683593750000159960937500001727578125000001140201562500000513090703125000016624138781250000398979330750000007181627953500000096951977372250000969519773722500006980542370802000034268117093028000102804351279084001423444863864240444480273022320640
1441503906250001045898437500001223701171875000881064843750000043612709765625000157005755156250000423915538921875000872054822925000000137348634610687500016481836153282500001483365253795425000970929984302460000436918492936107000120992813428460400155562188693734807556164641379450880
Technical details
Node id
n-9960d61ffc0b2713 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dca555f310728b4d

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:        20,         36;
  2:       250,        900,         810;
  3:      2500,      13500,       24300,        14580;
  4:     21875,     157500,      425250,       510300,       229635;
  5:    175000,    1575000,     5670000,     10206000,      9185400,       3306744;
  6:   1312500,   14175000,    63787500,    153090000,    206671500,     148803480,      44641044;
  7:   9375000,  118125000,   637875000,   1913625000,   3444525000,    3720087000,    2232052200,    573956280;
  8:  64453125,  928125000,  5847187500,  21049875000,  47362218750,   68201595000,   61381435500,  31567595400,   7102708965;
  9: 429687500, 6960937500, 50118750000, 210498750000, 568346625000, 1023023925000, 1227628710000, 947027862000, 426162537900, 85232507580;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+15+(9*n+27)*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[Together[((5*n + 15 + (9*n + 27)*x)*P[n - 1, x])/(n)]];
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 + 15 + (9*n + 27)*x)*v[i - 1])/(n)); 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.