Real-rooted atlas

A013622 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
155
28050130
396012003752535
4102401920012000250043940
510240025600024000010000015625714025
69830403072000384000024000007500009375011138790
79175040344064005376000044800000210000005250000546875168938315
8838860803670016006881280007168000004480000001680000003500000031250002509940680
975497472037748736008257536000103219200008064000000403200000012600000002250000001757812536707882445
106710886400377487360009437184000013762560000012902400000080640000000336000000009000000000140625000097656250530224968650
115905580032036909875200010380902400001730150400000189235200000014192640000007392000000002640000000006187500000085937500005371093757582217051695
125153960755203543348019200110729625600002076180480000025952256000000227082240000001419264000000063360000000001980000000000412500000000515625000002929687500107529623642220
1344667659878403350074490880011515881062400023991418880000033737932800000033737932800000024600576000000013178880000000051480000000000143000000000002681250000000304687500000158691406251514375532961265
143848290697216031267361914880011725260718080002687038914560000419849830400000047233105920000003936092160000000246005760000000011531520000000004004000000000001001000000000001706250000000017773437500008544921875021201257461457710
Technical details
Node id
n-dffda7b8609ca0a7 · revision 0 · JSON
Definition source
derived:derivative
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:         5;
  2:        80,         50;
  3:       960,       1200,        375;
  4:     10240,      19200,      12000,        2500;
  5:    102400,     256000,     240000,      100000,      15625;
  6:    983040,    3072000,    3840000,     2400000,     750000,      93750;
  7:   9175040,   34406400,   53760000,    44800000,   21000000,    5250000,     546875;
  8:  83886080,  367001600,  688128000,   716800000,  448000000,  168000000,   35000000,   3125000;
  9: 754974720, 3774873600, 8257536000, 10321920000, 8064000000, 4032000000, 1260000000, 225000000, 17578125;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (8*n+5*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 5*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] = 5; for(n = 2, N, my(i = n + 1); v[i] = ((8*n + 5*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.