Real-rooted atlas

A136572 → basis A013611 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(5\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
2508032162
3750180014403844374
4150004800057600307206144157464
53750001500000240000019200007680001228807085880
6112500005400000010800000011520000069120000221184002949120382637520
7393750000220500000052920000007056000000564480000027095040007225344008257536024106163760
8157500000001008000000002822400000004515840000004515840000002890137600001156055040002642411520026424115201735643790720
9708750000000510300000000016329600000000304819200000003657830400000029262643200000156067430400005350883328000107017666560095126814720140587147048320
10354375000000002835000000000001020600000000000217728000000000030481920000000002926264320000000195084288000000089181388800000026754416640000047563407360000380507258880012652843234348800
1119490625000000001715175000000000068607000000000000164656800000000000263450880000000000295064985600000000236051988480000000134886850560000000539547402240000001438793072640000023020689162240001674231939072001252631480200531200
12116943750000000000112266000000000000049397040000000000001317254400000000000023710579200000000000303495413760000000002832623861760000000019423706480640000000971185324032000000034531033743360000008287448098406400001205446996131840008036313307545600135284199861657369600
1376013437500000000007905397500000000000037945908000000000000011130799680000000000002226159936000000000000320567030784000000000034193816616960000000002735505329356800000000164130319761408000000072946808782848000000023342978810511360000050930135586570240000679068474487603200041788829199237120015828251383813912243200
145320940625000000000005959453500000000000000309891582000000000000009916530624000000000000021816367372800000000000034906187796480000000000041887425355776000000000038297074610995200000000026807952227696640000000014297574521438208000000057190298085752832000000166371776249462784000003327435524989255680000409530526152523776000234017443515727872001994359674360552942643200
Technical details
Node id
n-dbabf32666f7e9da · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6b726760bd05165a

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:            5,             4;
  2:           50,            80,             32;
  3:          750,          1800,           1440,            384;
  4:        15000,         48000,          57600,          30720,           6144;
  5:       375000,       1500000,        2400000,        1920000,         768000,         122880;
  6:     11250000,      54000000,      108000000,      115200000,       69120000,       22118400,        2949120;
  7:    393750000,    2205000000,     5292000000,     7056000000,     5644800000,     2709504000,      722534400,      82575360;
  8:  15750000000,  100800000000,   282240000000,   451584000000,   451584000000,   289013760000,   115605504000,   26424115200,    2642411520;
  9: 708750000000, 5103000000000, 16329600000000, 30481920000000, 36578304000000, 29262643200000, 15606743040000, 5350883328000, 1070176665600, 95126814720;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+4*n*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[(5*n + 4*n*x)*P[n - 1, x]];
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 + 4*n*x)*v[i - 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.