Real-rooted atlas

A331333: Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n.

A013609 → ⊙ 1/k!

A331333 nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A331333
\(\left(2\right)\,P_{n}(x) = \left(2 + 4\,x\right)\,P_{n-1}(x) + \left(1 + 4\,x\right)\,P_{n-1}'(x) + \left(x\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
228414
363636886
42419228812816648
5120120024001600400325752
672086402160019200720011526458576
7504070560211680235200117600282243136128671568
84032064512022579203010560188160060211210035281922568546432
9362880653184026127360406425603048192012192768270950433177620736512119401856
10362880072576000326592000580608000508032000243855360677376001105920010368005120010241815177984
113991680087816960043908480008781696000878169600049177497601639249920334540800418176003097600123904204829808908032
124790016001149603840063228211200140507136000158070528000101165137920393419980809634775040150543360014868480089210882949124096525586164736
1362270208001619025408009714152448002374570598400296821324800021371135385609498282393602713794969605088365568062819328005025546242492006469222481929898343691264
1487178291200244099215360015866448998400423105306624005817697966080046541583728640232707918643207598625914880166219941888024625176576024625176576162811084867837952160563216384198227905206272

Generated b-files: A087912 (row sums), A331333 ?

Computed in your browser ?

Technical details
Node id
n-8b66c9f49e6d2f0a · revision 1 · JSON
Definition source
oeis_catalog:A331333
Fingerprint
f6-9678f4e916f31c27

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): no

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      1,       2;
  2:      2,       8,        4;
  3:      6,      36,       36,        8;
  4:     24,     192,      288,      128,       16;
  5:    120,    1200,     2400,     1600,      400,       32;
  6:    720,    8640,    21600,    19200,     7200,     1152,      64;
  7:   5040,   70560,   211680,   235200,   117600,    28224,    3136,    128;
  8:  40320,  645120,  2257920,  3010560,  1881600,   602112,  100352,   8192,   256;
  9: 362880, 6531840, 26127360, 40642560, 30481920, 12192768, 2709504, 331776, 20736, 512;
%F Formula
The row polynomials satisfy 2*P(n,x) = (2+4*x)*P(n-1,x) + (1+4*x)*d/dx P(n-1,x) + x*d^2/dx^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[((2 + 4*x)*P[n - 1, x] + (1 + 4*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}])/(2)]];
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] = ((2 + 4*x)*v[i - 1] + (1 + 4*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x))/(2)); 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)

Generated with polytool c48629374597.

A331333: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      1,       2;
  2:      2,       8,        4;
  3:      6,      36,       36,        8;
  4:     24,     192,      288,      128,       16;
  5:    120,    1200,     2400,     1600,      400,       32;
  6:    720,    8640,    21600,    19200,     7200,     1152,      64;
  7:   5040,   70560,   211680,   235200,   117600,    28224,    3136,    128;
  8:  40320,  645120,  2257920,  3010560,  1881600,   602112,  100352,   8192,   256;
  9: 362880, 6531840, 26127360, 40642560, 30481920, 12192768, 2709504, 331776, 20736, 512;
%F Formula
The row polynomials satisfy 2*P(n,x) = (2+4*x)*P(n-1,x) + (1+4*x)*d/dx P(n-1,x) + x*d^2/dx^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[((2 + 4*x)*P[n - 1, x] + (1 + 4*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}])/(2)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = ((2 + 4*x)*v[i - 1] + (1 + 4*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x))/(2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-dee2eaa32f0764ef
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-dee2eaa32f0764ef
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-dee2eaa32f0764ef
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-dee2eaa32f0764ef

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [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
  5. [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
  6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.