Real-rooted atlas

A013622 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(64 + 80\,x + 25\,x^{2} + 40\,n + 25\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113518
214413025299
3147221609751254732
4143362944021600650062572501
51351683584003680001800004062531251085318
6124518440550405376000368000013500002437501562515965599
711272192435814407096320062720000322000009450000142187578125231686832
81006632964508876808716288009461760006272000002576000006300000081250003906253325671401
988919244845298483201014497280013074432000106444800005644800000193200000040500000045703125195312547312381818
10778462822444459622400113246208000169082880000163430400000106444800000470400000001380000000025312500002539062509765625668083460499
1167645734912428154552320122263961600020761804800002324889600000179773440000097574400000036960000000094875000000154687500001396484375488281259374377445732
125841155522564058744094720128446365696002445279232000031142707200000278986752000001797734400000083635200000002772000000000632500000000928125000007617187500244140625130827708764701
13501652180172837967510896640131909183078400278300459008000397357875200000404855193600000302235648000000166932480000000679536000000002002000000000041112500000005484375000004125976562512207031251817250639553518
14428809534832643511565261209601328862881382400307788093849600048702580326400005563010252800000472331059200000030223564800000001460659200000000528528000000000140140000000000261625000000003199218750000222167968750610351562525138633847156999
Technical details
Node id
n-3470a64b61130b24 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-354a4e768d29be07

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:        13,          5;
  2:       144,        130,          25;
  3:      1472,       2160,         975,         125;
  4:     14336,      29440,       21600,        6500,         625;
  5:    135168,     358400,      368000,      180000,       40625,       3125;
  6:   1245184,    4055040,     5376000,     3680000,     1350000,     243750,      15625;
  7:  11272192,   43581440,    70963200,    62720000,    32200000,    9450000,    1421875,     78125;
  8: 100663296,  450887680,   871628800,   946176000,   627200000,  257600000,   63000000,   8125000,   390625;
  9: 889192448, 4529848320, 10144972800, 13074432000, 10644480000, 5644800000, 1932000000, 405000000, 45703125, 1953125;
%F Formula
The row polynomials satisfy (5*n+3+5*x)*P(n,x) = (40*n+64+(25*n+80)*x+25*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[((40*n + 64 + (25*n + 80)*x + 25*x^2)*P[n - 1, x])/(5*n + 3 + 5*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] = ((40*n + 64 + (25*n + 80)*x + 25*x^2)*v[i - 1])/(5*n + 3 + 5*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

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.