Real-rooted atlas

A038231 → D → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
210414
3756012147
4500600240321372
53125500030008008012005
6187503750030000120002400192100842
7109375262500262500140000420006720448823543
86250001750000210000014000005600001344001792010246588344
935156251125000015750000126000006300000201600040320046080230451883209
1019531250703125001125000001050000006300000025200000672000011520001152005120403536070
1110742187542968750077343750082500000057750000027720000092400000211200003168000281600112643107227739
125859375002578125000515625000061875000004950000000277200000011088000003168000006336000084480006758402457623727920916
1331738281251523437500033515625000446875000004021875000025740000000120120000004118400000102960000018304000021964800159744053248179936733613
141708984375088867187500213281250000312812500000312812500000225225000000120120000000480480000001441440000032032000005125120005591040037273601146881356446145698
Technical details
Node id
n-4a35178dcacc3105 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-9629d1e9c04dbfbe

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:       1;
  2:      10,        4;
  3:      75,       60,       12;
  4:     500,      600,      240,       32;
  5:    3125,     5000,     3000,      800,      80;
  6:   18750,    37500,    30000,    12000,    2400,     192;
  7:  109375,   262500,   262500,   140000,   42000,    6720,    448;
  8:  625000,  1750000,  2100000,  1400000,  560000,  134400,  17920,  1024;
  9: 3515625, 11250000, 15750000, 12600000, 6300000, 2016000, 403200, 46080, 2304;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+2*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 2*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 2*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.