Real-rooted atlas

A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.

A156992 palindromic γ-positive nonnegative coefficients only rational roots

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

Recurrence from OEIS A156992
\(P_{n}(x) = \left(1 + x + n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2612624
324727224192
41204807204801201920
5720360072007200360072023040
65040302407560010080075600302405040322560
74032028224084672014112001411200846720282240403205160960
836288029030401016064020321280254016002032128010160640290304036288092897280
93628800326592001306368003048192004572288004572288003048192001306368003265920036288001857945600
1039916800399168000179625600047900160008382528000100590336008382528000479001600017962560003991680003991680040874803200
114790016005269017600263450880007903526400015807052800022129873920022129873920015807052800079035264000263450880005269017600479001600980995276800
12622702080074724249600410983372800136994457600030823752960004931800473600575376721920049318004736003082375296000136994457600041098337280074724249600622702080025505877196800
138717829120011333177856006799906713600249329912832006233247820800011219846077440014959794769920014959794769920011219846077440062332478208000249329912832006799906713600113331778560087178291200714164561510400
14130767436800018307441152000118998367488000475993469952000130898204236800026179640847360003926946127104000448793843097600039269461271040002617964084736000130898204236800047599346995200011899836748800018307441152000130767436800021424936845312000

Generated b-files: A156992, A156992 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-75a6475b3e851dc5 · revision 1 · JSON
Definition source
oeis_catalog:A156992
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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:       2,        2;
  2:       6,       12,         6;
  3:      24,       72,        72,        24;
  4:     120,      480,       720,       480,       120;
  5:     720,     3600,      7200,      7200,      3600,       720;
  6:    5040,    30240,     75600,    100800,     75600,     30240,      5040;
  7:   40320,   282240,    846720,   1411200,   1411200,    846720,    282240,     40320;
  8:  362880,  2903040,  10160640,  20321280,  25401600,  20321280,  10160640,   2903040,   362880;
  9: 3628800, 32659200, 130636800, 304819200, 457228800, 457228800, 304819200, 130636800, 32659200, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (n+1+(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 + 1 + (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 + 1 + (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.

A156992: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:       1;
   2:       2,        2;
   3:       6,       12,         6;
   4:      24,       72,        72,        24;
   5:     120,      480,       720,       480,       120;
   6:     720,     3600,      7200,      7200,      3600,       720;
   7:    5040,    30240,     75600,    100800,     75600,     30240,      5040;
   8:   40320,   282240,    846720,   1411200,   1411200,    846720,    282240,     40320;
   9:  362880,  2903040,  10160640,  20321280,  25401600,  20321280,  10160640,   2903040,   362880;
  10: 3628800, 32659200, 130636800, 304819200, 457228800, 457228800, 304819200, 130636800, 32659200, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (n+n*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 + n*x)*P[n - 1, x]];
Table[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 + n*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A156992: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:       1;
   2:       2,        2;
   3:       6,       12,         6;
   4:      24,       72,        72,        24;
   5:     120,      480,       720,       480,       120;
   6:     720,     3600,      7200,      7200,      3600,       720;
   7:    5040,    30240,     75600,    100800,     75600,     30240,      5040;
   8:   40320,   282240,    846720,   1411200,   1411200,    846720,    282240,     40320;
   9:  362880,  2903040,  10160640,  20321280,  25401600,  20321280,  10160640,   2903040,   362880;
  10: 3628800, 32659200, 130636800, 304819200, 457228800, 457228800, 304819200, 130636800, 32659200, 3628800;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n+n*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 + n*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 + n*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.

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-d07da0cb5a6ac36b
  • 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-d07da0cb5a6ac36b, e-e811f0fabc9e8e88
  • 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-d07da0cb5a6ac36b, e-e811f0fabc9e8e88
  • 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-d07da0cb5a6ac36b
  • 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-e811f0fabc9e8e88

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 8 proof steps

Same fingerprint (59)

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

References

  1. [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