Real-rooted atlas

A147716 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(15\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
212032152
3270014401924332
45400043200115201024109744
5101250010800004320007680051202606420
618225000243000001296000034560004608002457659426376
73189375005103000003402000001209600002419200025804801146881317284668
8546750000010206000000816480000036288000009676800001548288001376256052428828603895648
99226406250019683000000018370800000097977600000326592000006967296000928972800707788802359296611408269476
1015377343750003690562500000393660000000024494400000009797760000002612736000004644864000053084160003538944001048576012907507911160
1125372617187500676603125000008119237500000057736800000000269438400000008622028800000191600640000029196288000029196288000173015040046137344269766915343244
124151882812500001217885625000000162384750000000012990780000000006928416000000002586608640000006897623040000013138329600000175177728000015571353600083047219202013265925591532427114512
1367468095703125002158979062500000031665026250000000281466900000000001688801400000000072055526400000002241727488000000512394854400000853991424000001012137984000080971038720039258685440872415232115092375791440372
141089869238281250003778213359375000006045141375000000005910804900000000003940536600000000001891457568000000006725182464000000017933819904000000358676398080000053137244160000056679727104000412216197120018320719872037580963842354967073886395304
Technical details
Node id
n-6be72d7a1f4aa929 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-1bc3b9efcd292160

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:           4;
  2:         120,           32;
  3:        2700,         1440,          192;
  4:       54000,        43200,        11520,        1024;
  5:     1012500,      1080000,       432000,       76800,        5120;
  6:    18225000,     24300000,     12960000,     3456000,      460800,      24576;
  7:   318937500,    510300000,    340200000,   120960000,    24192000,    2580480,    114688;
  8:  5467500000,  10206000000,   8164800000,  3628800000,   967680000,  154828800,  13762560,   524288;
  9: 92264062500, 196830000000, 183708000000, 97977600000, 32659200000, 6967296000, 928972800, 70778880, 2359296;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (15*n+4*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((15*n + 4*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((15*n + 4*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.