Real-rooted atlas

A269944: Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.

A269944 nonnegative coefficients only rational roots

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

Recurrence from OEIS A269944
\(P_{n}(x) = \left(1 + x - 2\,n + n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20112
3045110
403649141100
505768202733011700
6014400210767645102355144200
705184007731362962964447330039111635400

Show all 15 stored rows

Generated b-files: A008955 (reversed triangle), A101686 (row sums), A269944 ?

Computed in your browser ?

Technical details
Node id
n-54284d24a094e2d3 · revision 1 · JSON
Definition source
oeis_catalog:A269944
Fingerprint
f6-d0a612563fb7dda7

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: 0,          1;
  2: 0,          1,          1;
  3: 0,          4,          5,          1;
  4: 0,         36,         49,         14,         1;
  5: 0,        576,        820,        273,        30,        1;
  6: 0,      14400,      21076,       7645,      1023,       55,      1;
  7: 0,     518400,     773136,     296296,     44473,     3003,     91,     1;
  8: 0,   25401600,   38402064,   15291640,   2475473,   191620,   7462,   140,   1;
  9: 0, 1625702400, 2483133696, 1017067024, 173721912, 14739153, 669188, 16422, 204, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n^2-2*n+1+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^2 - 2*n + 1 + 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^2 - 2*n + 1 + 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.

A008955: reversed triangle, offset 0, from the row P1, low zeros dropped
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   5,     4;
  3: 1,  14,    49,      36;
  4: 1,  30,   273,     820,      576;
  5: 1,  55,  1023,    7645,    21076,      14400;
  6: 1,  91,  3003,   44473,   296296,     773136,      518400;
  7: 1, 140,  7462,  191620,  2475473,   15291640,    38402064,    25401600;
  8: 1, 204, 16422,  669188, 14739153,  173721912,  1017067024,  2483133696,   1625702400;
  9: 1, 285, 32946, 1999370, 68943381, 1367593305, 15088541896, 84865562640, 202759531776, 131681894400;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = (n^2+x)*P(n-1,x) for n >= 1, with P(0,x) = x.
%t Mathematica
P[0, x_] := x;
P[n_, x_] := P[n, x] = Expand[(n^2 + x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; for(n = 1, N, my(i = n + 1); v[i] = (n^2 + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A269944: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 0,          1;
  2: 0,          1,          1;
  3: 0,          4,          5,          1;
  4: 0,         36,         49,         14,         1;
  5: 0,        576,        820,        273,        30,        1;
  6: 0,      14400,      21076,       7645,      1023,       55,      1;
  7: 0,     518400,     773136,     296296,     44473,     3003,     91,     1;
  8: 0,   25401600,   38402064,   15291640,   2475473,   191620,   7462,   140,   1;
  9: 0, 1625702400, 2483133696, 1017067024, 173721912, 14739153, 669188, 16422, 204, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n^2-2*n+1+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^2 - 2*n + 1 + 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^2 - 2*n + 1 + 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-aa0ec4b5da096d18
  • 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-aa0ec4b5da096d18
  • 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-aa0ec4b5da096d18

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

Used in 5 proof steps

Same fingerprint (1)

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