Real-rooted atlas

A049020: Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.

A023531 → basis A049020

A049020 nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A049020
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(1 + x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
22316
35106122
415373110194
55215116075151454
62036748565201552112430
7877326348023556140028728114214
8414017007283372462611991329049036189918
92114794828175896174805101031346716972786451610182
1011597556259511469311279240853315350889889771362012005514412798
1167857035350277841108967715173002603492511106906820753724915176066133827974
1242135972343084056089804757507526364100634669734124287462928684447612431752497781273646526
13276444371632548854189525086136568365673044523462318861421717443943911473457679043327150034459112326980998
141908993221192059223326108291751427976605181338526349772623216175078925164507921136900051710408717287271139324641105120732504062

Generated b-files: A001861 (row sums), A049020 ?

Computed in your browser ?

Technical details
Node id
n-c15181c2ebe13c1d · revision 2 · JSON
Definition source
oeis_catalog:A049020
Fingerprint
f6-fdf6b18bd9ec2fbc

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,     1;
  2:     2,     3,      1;
  3:     5,    10,      6,      1;
  4:    15,    37,     31,     10,      1;
  5:    52,   151,    160,     75,     15,     1;
  6:   203,   674,    856,    520,    155,    21,    1;
  7:   877,  3263,   4802,   3556,   1400,   287,   28,   1;
  8:  4140, 17007,  28337,  24626,  11991,  3290,  490,  36,  1;
  9: 21147, 94828, 175896, 174805, 101031, 34671, 6972, 786, 45, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (1+x)*d/dx 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[(1 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], 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] = (1 + x)*v[i - 1] + (1 + x)*deriv(v[i - 1], 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)

Generated with polytool c48629374597.

A049020: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     1,     1;
  2:     2,     3,      1;
  3:     5,    10,      6,      1;
  4:    15,    37,     31,     10,      1;
  5:    52,   151,    160,     75,     15,     1;
  6:   203,   674,    856,    520,    155,    21,    1;
  7:   877,  3263,   4802,   3556,   1400,   287,   28,   1;
  8:  4140, 17007,  28337,  24626,  11991,  3290,  490,  36,  1;
  9: 21147, 94828, 175896, 174805, 101031, 34671, 6972, 786, 45, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (1+x)*d/dx 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[(1 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];
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] = (1 + x)*v[i - 1] + (1 + x)*deriv(v[i - 1], x)); 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, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-2ff0affdc4d5d50d, e-bcbfbcbb753a93dd
  • 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-2ff0affdc4d5d50d, e-bcbfbcbb753a93dd
  • 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-2ff0affdc4d5d50d, e-bcbfbcbb753a93dd

Proof chains

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

Used in 5 proof steps

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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. [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
  4. [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
  5. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741