Real-rooted atlas

A013622 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(8\,n + 10\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 10\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11010
2160200360
31920480030009720
420480768009600040000233280
52048001024000192000016000005000005248800
61966080122880003072000038400000240000006000000113374080
718350080137625600430080000716800000672000000336000000700000002380855680
816777216014680064005505024000114688000001433600000010752000000448000000080000000048977602560
915099494401509949440066060288000165150720000258048000000258048000000161280000000576000000009000000000991796451840
10134217728001509949440007549747200002202009600000412876800000051609600000004300800000000230400000000072000000000010000000000019835929036800
111181116006401476395008000830472192000027682406400000605552640000009083289600000094617600000000675840000000003168000000000088000000000001100000000000392751394928640
12103079215104014173392076800885837004800003321888768000008304721920000001453326336000000181665792000000016220160000000001013760000000000422400000000000105600000000000120000000000007712209209507840
138933531975680134002979635200921270484992000383862702080000010796138496000000215922769920000003148873728000000033737932800000000263577600000000001464320000000000054912000000000001248000000000000130000000000000150388079585402880
14769658139443201250694476595200938020857446400042992622632960000134351945728000000302291877888000000503819796480000000629774745600000000590413824000000000410009600000000000205004800000000000698880000000000001456000000000000014000000000000002915215081193963520
Technical details
Node id
n-7ad950efd156275d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6b726760bd05165a

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:         10;
  2:        160,         200;
  3:       1920,        4800,        3000;
  4:      20480,       76800,       96000,        40000;
  5:     204800,     1024000,     1920000,      1600000,       500000;
  6:    1966080,    12288000,    30720000,     38400000,     24000000,      6000000;
  7:   18350080,   137625600,   430080000,    716800000,    672000000,    336000000,     70000000;
  8:  167772160,  1468006400,  5505024000,  11468800000,  14336000000,  10752000000,   4480000000,   800000000;
  9: 1509949440, 15099494400, 66060288000, 165150720000, 258048000000, 258048000000, 161280000000, 57600000000, 9000000000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (8*n+10*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 10.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 10;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 10*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] = 10; for(n = 2, N, my(i = n + 1); v[i] = ((8*n + 10*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.