Real-rooted atlas

A130534: Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.

A130534 nonnegative coefficients only rational roots

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

Recurrence from OEIS A130534
\(P_{n}(x) = \left(x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
22316
36116124
4245035101120
512027422585151720
6720176416247351752115040
7504013068131326769196032228140320
84032010958411812467284224494536546361362880
9362880102657611727007236802693256327394508704513628800
10362880010628640127535768409500341693090205515777318150132055139916800
113991680012054384015091797610525807645995730133395352637558357423326701925661479001600
124790016001486442880193155955214140148886572068362060701504499023169266347494635577027177816227020800
1362270208001980275904026596717056203137530969957703756333611878679094315313503647316669653147447391091373191187178291200
148717829120028346564736039215679782431098926040015972160568056663366760144093229282681453775368411615373122752749747143325500510511307674368000

Generated b-files: A094638 (reversed triangle), A130534 ?

Computed in your browser ?

Technical details
Node id
n-efe23a76953cf69d · revision 3 · JSON
Definition source
oeis_catalog:A130534
Fingerprint
f6-34eaf45657600f7a

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

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:      6,      11,       6,      1;
  4:     24,      50,      35,     10,      1;
  5:    120,     274,     225,     85,     15,     1;
  6:    720,    1764,    1624,    735,    175,    21,    1;
  7:   5040,   13068,   13132,   6769,   1960,   322,   28,   1;
  8:  40320,  109584,  118124,  67284,  22449,  4536,  546,  36,  1;
  9: 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+x)*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[(n + x)*P[n - 1, 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] = (n + x)*v[i - 1]); 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.

A094638: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,  1;
   3: 1,  3,   2;
   4: 1,  6,  11,    6;
   5: 1, 10,  35,   50,    24;
   6: 1, 15,  85,  225,   274,    120;
   7: 1, 21, 175,  735,  1624,   1764,    720;
   8: 1, 28, 322, 1960,  6769,  13132,  13068,    5040;
   9: 1, 36, 546, 4536, 22449,  67284, 118124,  109584,   40320;
  10: 1, 45, 870, 9450, 63273, 269325, 723680, 1172700, 1026576, 362880;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n-1+x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n - 1 + x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (n - 1 + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A130534: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      1,       1;
  2:      2,       3,       1;
  3:      6,      11,       6,      1;
  4:     24,      50,      35,     10,      1;
  5:    120,     274,     225,     85,     15,     1;
  6:    720,    1764,    1624,    735,    175,    21,    1;
  7:   5040,   13068,   13132,   6769,   1960,   322,   28,   1;
  8:  40320,  109584,  118124,  67284,  22449,  4536,  546,  36,  1;
  9: 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+x)*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[(n + x)*P[n - 1, 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] = (n + x)*v[i - 1]); 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-5f21311bb67f8c1c
  • 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-0b3ba9047fae59ed, e-1ba7a8eebd893648, e-2e5c10abc86c264d, e-5f21311bb67f8c1c, e-90e69a839bdbf74b
  • 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-0b3ba9047fae59ed, e-1ba7a8eebd893648, e-2e5c10abc86c264d, e-5f21311bb67f8c1c, e-90e69a839bdbf74b
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-5f21311bb67f8c1c
  • 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-0b3ba9047fae59ed, e-2e5c10abc86c264d, e-90e69a839bdbf74b
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, 0]: (-inf, 0] -> (-inf, 0] through R_s, s = 1, and U_g
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-d61ef81b134a745f
  • 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-1ba7a8eebd893648

Proof chains

6 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 17 proof steps

Same fingerprint (6)

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

References

  1. [Bre89] Francesco Brenti. Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc., 81(413):viii+106, 1989. doi:10.1090/memo/0413
  2. [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
  3. [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