Real-rooted atlas

A178821 → basis A013610 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(20 + 48\,x + 5\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1256085
2375180021604335
34375315007560060480171955
4437504200001512000241920014515205846470
5393750472500022680000544320006531840031352832178901982
6328125047250000283500000907200000163296000015676416006270566405068889490
72578125043312500031185000001247400000029937600000431101440003448811520011824496640135411762090
819335937537125000003118500000014968800000044906400000086220288000010346434560007094697984002128409395203452999933295
91396484375301640625002895750000001621620000000583783200000014010796800000224172748800002305776844800013834661068800368924295168084790331695355
10977539062523460937500025337812500001621620000000068108040000000196151155200000392302310400000538014597120000484213137408000258247006617600619792815882242018009894349449
116665039062517595703125002111484375000015202687500000072972900000000024518894400000005884534656000000100877736960000001210532843520000096842627481600004648446119116800101420642598912046781138459919045
124443359375001279687500000016891875000000013513500000000007297290000000000280215936000000007846046208000000016140437913600000024210656870400000025824700661760000018593784476467200081136514079129600162273028158259201060372471758165020
132905273437500906445312500001305281250000000114864750000000006891885000000000029772943200000000095273418240000000022865620377600000004115811667968000000548774889062400000052682389349990400003448301848363008000137932073934520320025464382880219136023572895718316130060
141867675781250062753906250000097896093750000009398025000000000062026965000000000029772943200000000001071825955200000000029398654771200000000617371750195200000009877948003123200000011853537603747840000010344905545089024000062069433270534144000229179445921972224003928790501519523840515236149271766842740
Technical details
Node id
n-84f28256d09faa74 · revision 0 · JSON
Definition source
derived:basis_map_A013611
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:         25,          60;
  2:        375,        1800,         2160;
  3:       4375,       31500,        75600,         60480;
  4:      43750,      420000,      1512000,       2419200,       1451520;
  5:     393750,     4725000,     22680000,      54432000,      65318400,       31352832;
  6:    3281250,    47250000,    283500000,     907200000,    1632960000,     1567641600,      627056640;
  7:   25781250,   433125000,   3118500000,   12474000000,   29937600000,    43110144000,    34488115200,    11824496640;
  8:  193359375,  3712500000,  31185000000,  149688000000,  449064000000,   862202880000,  1034643456000,   709469798400,   212840939520;
  9: 1396484375, 30164062500, 289575000000, 1621620000000, 5837832000000, 14010796800000, 22417274880000, 23057768448000, 13834661068800, 3689242951680;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+20+(12*n+48)*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 + 20 + (12*n + 48)*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 + 20 + (12*n + 48)*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.