Real-rooted atlas

A013611 → D → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
24064104
33009607682028
42000960015360819235152
5125008000019200020480081920571220
6750006000001920000307200024576007864328911032
74375004200000168000003584000043008000275251207340032135150652
8250000028000000134400000358400000573440000550502400293601280671088642007952544
91406250018000000010080000003225600000645120000082575360006606028800301989888060397977629366305956
107812500011250000007200000000268800000006451200000010321920000011010048000075497472000301989888005368709120424179974920
11429687500687500000049500000000211200000000591360000000113541120000015138816000001384120320000830472192000295279001600472446402566065773641356
1223437500004125000000033000000000015840000000005068800000000113541120000001816657920000020761804800000166094438400008858370048000283467841536041231686041686023698913776
131269531250024375000000021450000000001144000000000041184000000000105431040000000196804608000000269903462400000269903462400000191931351040000921270484992002680059592704035734127902721211500426369012
146835937500014218750000001365000000000080080000000000320320000000000922521600000000196804608000000031488737280000003778648473600000335879864320000021496311316480009380208574464002501388953190403078632557772816961005969166168
Technical details
Node id
n-09968aa1d1f31352 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-4b28493176e02e0b

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:       40,        64;
  3:      300,       960,        768;
  4:     2000,      9600,      15360,       8192;
  5:    12500,     80000,     192000,     204800,      81920;
  6:    75000,    600000,    1920000,    3072000,    2457600,     786432;
  7:   437500,   4200000,   16800000,   35840000,   43008000,   27525120,    7340032;
  8:  2500000,  28000000,  134400000,  358400000,  573440000,  550502400,  293601280,   67108864;
  9: 14062500, 180000000, 1008000000, 3225600000, 6451200000, 8257536000, 6606028800, 3019898880, 603979776;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+8*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[((5*n + 8*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] = ((5*n + 8*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.