Real-rooted atlas

A038303 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 4\,x + 4\,n\right)\,P_{n}(x) = \left(121 + 88\,x + 16\,x^{2} + 44\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115419
220912016345
327832508720646075
43593744528200643840256104625
54538717187404452801337601920010241771875
65636785108929048624880356224080256092160409629615625
7690908791578299801525006568049888024935680449433643008016384489796875
8837948353221090812825252796801626673664643991040159588352239697921966080655368030390625
9100748674073016614070839796346304303033561601464006297646367354889575301121232732168847360262144130707421875
101202553322414029946962806033228141605306179507203030335616001171205038083091156992054716006406163660803932160010485762114384765625
11142655835305552912346186048865883318160884873460768058367974579202666695342080858883694592194301296640300938035203013345280173015040419430434022373046875
1216833388566049684748009466401269896308464961418541330905601061848152921605603325559603221333562736640588948819148811658077798401605002854401446405734475497472016777216544934619140625
131977209877334238753362054345481780344824612640220115360133926418441037301772801104322079038464485621548498944158477894615040382816732446726735778283520834601484288683755438083271557120671088648693004638671875
1423130217136433771107237531307168824509413752167344332331033927692803081615041874969620653961777985536103070060710256643884972387991552110934526230528023819707796684837720358387712424888028364831908587110414092861440268435456138179849853515625
Technical details
Node id
n-dc8b18ad1be551e4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-6e35560835d8bd96

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:           1;
  1:          15,           4;
  2:         209,         120,          16;
  3:        2783,        2508,         720,          64;
  4:       35937,       44528,       20064,        3840,         256;
  5:      453871,      718740,      445280,      133760,       19200,       1024;
  6:     5636785,    10892904,     8624880,     3562240,      802560,      92160,      4096;
  7:    69090879,   157829980,   152500656,    80498880,    24935680,    4494336,    430080,     16384;
  8:   837948353,  2210908128,  2525279680,  1626673664,   643991040,  159588352,  23969792,   1966080,   65536;
  9: 10074867407, 30166140708, 39796346304, 30303356160, 14640062976, 4636735488, 957530112, 123273216, 8847360, 262144;
%F Formula
The row polynomials satisfy (4*n+7+4*x)*P(n,x) = (44*n+121+(16*n+88)*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((44*n + 121 + (16*n + 88)*x + 16*x^2)*P[n - 1, x])/(4*n + 7 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((44*n + 121 + (16*n + 88)*x + 16*x^2)*v[i - 1])/(4*n + 7 + 4*x)); 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

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.