Real-rooted atlas

A269947: Triangle read by rows, Stirling cycle numbers of order 3, 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)^3*T(n-1,k), for n>=0 and 0<=k<=n.

A269947 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 A269947
\(P_{n}(x) = \left(-1 + x + 3\,n - 3\,n^{2} + n^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1011
20112
3089118
40216251361504
5013824162802555100132760
60172800020488243356551505522514127760
70373248000444273984745503043587535636554411895723920
80128024064000152759224512260150282561305074809254212002149187841308129028480
906554832076800078340747014144134724536915846942133304641432072920913545921661632612961158070191610240
1004778472583987200057175952894078976989975948817888051955397159984011134024923825113070497673584760870156111020251115391239875475200
11047784725839872000000572237376199188480009956935441072958976529453731088018880116535788954248401242045225968256978313676732145870870358611030251115506631115350675200
12063601470092869632000000762125794979518586880001330990480968802724505671465985151922608825616040367240898480920176969798471798915105301807296958835539854956436918983280761238543561153854832645647099366400
130109903340320478724096000000131758938842553681444864000230757280906388629381447681248242128234910707751424284324144437918011180163218441790001670060401996585028563246979719430500944069215509988603483200731845601513955360841266015005644323834804505600
140241457638684091756838912000000289584291977410916858462208000508291335539761355565489192962765463683822737687868023296637142566612454977640325767355240757071587133878964708341486753620618903178024731343044473034126974997129284359610775081803533347825012850610182811584700982406223788900303308800

Generated b-files: A249677 (reversed triangle), A255433 (row sums), A269947 ?

Computed in your browser ?

Technical details
Node id
n-fefc87d068bda822 · revision 1 · JSON
Definition source
oeis_catalog:A269947
Fingerprint
f6-53af44bcc1c9acce

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,              8,              9,              1;
  4: 0,            216,            251,             36,            1;
  5: 0,          13824,          16280,           2555,          100,           1;
  6: 0,        1728000,        2048824,         335655,        15055,         225,         1;
  7: 0,      373248000,      444273984,       74550304,      3587535,       63655,       441,      1;
  8: 0,   128024064000,   152759224512,    26015028256,   1305074809,    25421200,    214918,    784,    1;
  9: 0, 65548320768000, 78340747014144, 13472453691584, 694213330464, 14320729209, 135459216, 616326, 1296, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*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^3 - 3*n^2 + 3*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^3 - 3*n^2 + 3*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.

A249677: reversed triangle, offset 0, from the row P1, low zeros dropped
%e Example
Triangle begins:
  0: 1;
  1: 1,    1;
  2: 1,    9,       8;
  3: 1,   36,     251,       216;
  4: 1,  100,    2555,     16280,        13824;
  5: 1,  225,   15055,    335655,      2048824,        1728000;
  6: 1,  441,   63655,   3587535,     74550304,      444273984,       373248000;
  7: 1,  784,  214918,  25421200,   1305074809,    26015028256,    152759224512,     128024064000;
  8: 1, 1296,  616326, 135459216,  14320729209,   694213330464,  13472453691584,   78340747014144,    65548320768000;
  9: 1, 2025, 1561110, 584760870, 113070497673, 11134024923825, 519553971599840, 9899759488178880, 57175952894078976, 47784725839872000;
%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^3+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^3 + 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^3 + 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.

A269947: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 0,              1;
  2: 0,              1,              1;
  3: 0,              8,              9,              1;
  4: 0,            216,            251,             36,            1;
  5: 0,          13824,          16280,           2555,          100,           1;
  6: 0,        1728000,        2048824,         335655,        15055,         225,         1;
  7: 0,      373248000,      444273984,       74550304,      3587535,       63655,       441,      1;
  8: 0,   128024064000,   152759224512,    26015028256,   1305074809,    25421200,    214918,    784,    1;
  9: 0, 65548320768000, 78340747014144, 13472453691584, 694213330464, 14320729209, 135459216, 616326, 1296, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*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^3 - 3*n^2 + 3*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^3 - 3*n^2 + 3*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-f7365f1dbad216ac
  • 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-f7365f1dbad216ac
  • 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-f7365f1dbad216ac

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

No candidate equalities or related sequences.