Real-rooted atlas

A178821 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(20 + 24\,x + 5\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1253055
23759005401815
343751575018900756046585
443750210000378000302400907201024870
5393750236250056700006804000408240097977620292426
632812502362500070875000113400000102060000489888009797760372027810
725781250216562500779625000155925000018711000001347192000538876800923788806430766430
819335937518562500007796250000187110000002806650000026943840000161663040005542732800831409920106107646095
9139648437515082031250723937500002027025000003648645000004378374000003502699200001801388160005404164480072055526401685932599065
1097753906251173046875006334453125002027025000000425675250000061297236000006129723600000420323904000018914575680005043886848006052664217625963362025601
11666503906258797851562505278710937500190033593750004560806250000076621545000000919458540000007881073200000047286439200000189145756800004539498163200495217981440389450430384015
12444335937500639843750000042229687500000168918750000000456080625000000875674800000000122594472000000012609717120000009457287840000005043886848000001815799265280003961743851520039617438515205711939645632220
1329052734375004532226562500032632031250000014358093750000004307428125000000930404475000000014886471600000000178637659200000001607738932800000010718259552000000514476458496000016837411368960003367482273792003108445175808082164054902555780
14186767578125003137695312500002447402343750000117475312500000003876685312500000093040447500000000167472805500000000229676990400000000241160839920000000192928671936000000115757203161600000505122341068800001515367023206400027976006582272002397943421337601162034490764717460
Technical details
Node id
n-31672dcf78468367 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-da2debf48cb6165b

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,          30;
  2:        375,         900,         540;
  3:       4375,       15750,       18900,         7560;
  4:      43750,      210000,      378000,       302400,        90720;
  5:     393750,     2362500,     5670000,      6804000,      4082400,       979776;
  6:    3281250,    23625000,    70875000,    113400000,    102060000,     48988800,      9797760;
  7:   25781250,   216562500,   779625000,   1559250000,   1871100000,   1347192000,    538876800,     92378880;
  8:  193359375,  1856250000,  7796250000,  18711000000,  28066500000,  26943840000,  16166304000,   5542732800,   831409920;
  9: 1396484375, 15082031250, 72393750000, 202702500000, 364864500000, 437837400000, 350269920000, 180138816000, 54041644800, 7205552640;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n+20+(6*n+24)*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 + (6*n + 24)*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 + (6*n + 24)*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.