Real-rooted atlas

A013611 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 16\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11616
2160512672
3120076801228821168
4800076800245760262144592704
55000064000030720006553600524288015558480
630000048000003072000098304000157286400100663296392073696
717500003360000026880000011468800002752512000352321536018790481929605805552
81000000022400000021504000001146880000036700160000704643072007516192768034359738368230539333248
9562500001440000000161280000001032192000004128768000001056964608000169114337280015461882265606184752906245446491747984
10312500000900000000011520000000086016000000041287680000001321205760000028185722880000386547056640003092376453120010995116277760127084807452960
111718750000550000000007920000000006758400000000378470400000001453326336000003875536896000007086696038400008504035246080006047313952768001935140464885762935659052163376
1293750000003300000000005280000000000506880000000003244032000000001453326336000000465064427520000010630044057600000170080704921600001814194185830400011610842789314560337769972052787267253280104106432
135078125000019500000000003432000000000036608000000000026357760000000001349517312000000050381979648000000138190572748800000276381145497600000393075406929920000377352390652723200219550481834311680585467951558164481530012122368421328
1427343750000011375000000000218400000000000256256000000000020500480000000000118082764800000000503819796480000000161222334873600000038693360369664000006878819621273600000880488911523020800076842668642009088004098275660907151360100880631653099110434601812613562759264
Technical details
Node id
n-d2150def566f2478 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-833b492313a93e6e

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:        0;
  1:       16;
  2:      160,        512;
  3:     1200,       7680,       12288;
  4:     8000,      76800,      245760,       262144;
  5:    50000,     640000,     3072000,      6553600,      5242880;
  6:   300000,    4800000,    30720000,     98304000,    157286400,     100663296;
  7:  1750000,   33600000,   268800000,   1146880000,   2752512000,    3523215360,    1879048192;
  8: 10000000,  224000000,  2150400000,  11468800000,  36700160000,   70464307200,   75161927680,   34359738368;
  9: 56250000, 1440000000, 16128000000, 103219200000, 412876800000, 1056964608000, 1691143372800, 1546188226560, 618475290624;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+16*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 16.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 16;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 16*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 16; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 16*n*x)*v[i - 1])/(n - 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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.