Real-rooted atlas

A013622 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-3 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(16 + 80\,x + 100\,x^{2} + 40\,n + 100\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1282048
238411204001904
343522304033600800068992
4450563481609216008960001600002370816
54423684505600174080003072000022400000320000078675968
6419430453084160270336000696320000921600000537600000640000002547134464
738797312587202560371589120012615680000243712000002580480000012544000000128000000080957571072
8352321536620756992046976204800198180864000504627200000779878400000688128000000286720000000256000000002536670560256
931541166086341787648055868129280028185722880008918138880000181665792000002339635200000017694720000000645120000000051200000000078582815653888
1027917287424630823321600634178764800037245419520000140928614400000356725555200000605552640000000668467200000000442368000000000143360000000000102400000000002411887957377024
112448131358726141803233280693905653760004650644275200002048498073600000620085903360000013079937024000000190316544000000001838284800000000010813440000000000315392000000000020480000000000073456798140465152
12213030377881658755152609280737016387993600555124523008000027903865651200000983279075328000002480343613440000004484549836800000005709496320000000004902092800000000002595225600000000006881280000000000040960000000000002222660537282461696
1318416819765248553878982492160763816983920640063874753626112000360830939955200000145100101386240000042608759930880000009212704849920000000145747869696000000001649410048000000000012745441280000000000613416960000000000014909440000000000008192000000000000066878860345693175808
14158329674399744515670953426944077543057548902400712895851659264000447123275382784000020206532637491200000677133806469120000001704350397235200000003224446697472000000004534378168320000000004618348134400000000003244294144000000000001431306240000000000003211264000000000000016384000000000000002002650318129367875584
Technical details
Node id
n-85e06efa45cc705a · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1644f7e5ece2752e

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:         28,          20;
  2:        384,        1120,          400;
  3:       4352,       23040,        33600,          8000;
  4:      45056,      348160,       921600,        896000,        160000;
  5:     442368,     4505600,     17408000,      30720000,      22400000,        3200000;
  6:    4194304,    53084160,    270336000,     696320000,     921600000,      537600000,       64000000;
  7:   38797312,   587202560,   3715891200,   12615680000,   24371200000,    25804800000,    12544000000,     1280000000;
  8:  352321536,  6207569920,  46976204800,  198180864000,  504627200000,   779878400000,   688128000000,   286720000000,   25600000000;
  9: 3154116608, 63417876480, 558681292800, 2818572288000, 8918138880000, 18166579200000, 23396352000000, 17694720000000, 6451200000000, 512000000000;
%F Formula
The row polynomials satisfy (5*n-3+5*x)*P(n,x) = (40*n+16+(100*n+80)*x+100*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 + 16 + (100*n + 80)*x + 100*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 + 16 + (100*n + 80)*x + 100*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.