Real-rooted atlas

A145324: Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....

A145324 nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0) · all roots real, in [−1, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A145324
\(P_{n}(x) = \left(1 + x + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
215612
319262460
411471154120360
512015558010447202520
6127295166551048028504020160
7135511402518424488606926440320181440

Show all 15 stored rows

Generated b-files: A136124 (reversed triangle), A143491 (reversed triangle), A145324 ?

Computed in your browser ?

Other recurrences

X ↦ x/(x+1) substitution · proved equal for all n (proof step)

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} + n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-2feb6dc01f424b0a · revision 2 · JSON
Definition source
oeis_catalog:A145324
derived:mobius_x_over_x_plus_one (equal by e-41f5b6f963fcfcc0)
Fingerprint
f6-2ee8a94f56aa5e4f

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,  2;
  2: 1,  5,    6;
  3: 1,  9,   26,    24;
  4: 1, 14,   71,   154,    120;
  5: 1, 20,  155,   580,   1044,    720;
  6: 1, 27,  295,  1665,   5104,   8028,    5040;
  7: 1, 35,  511,  4025,  18424,  48860,   69264,   40320;
  8: 1, 44,  826,  8624,  54649, 214676,  509004,  663696,  362880;
  9: 1, 54, 1266, 16884, 140889, 761166, 2655764, 5753736, 6999840, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (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[(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] = (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.

A136124: reversed triangle, offset 2, from the row P0
%e Example
Triangle begins:
   2:       1;
   3:       2,       1;
   4:       6,       5,       1;
   5:      24,      26,       9,       1;
   6:     120,     154,      71,      14,      1;
   7:     720,    1044,     580,     155,     20,      1;
   8:    5040,    8028,    5104,    1665,    295,     27,     1;
   9:   40320,   69264,   48860,   18424,   4025,    511,    35,    1;
  10:  362880,  663696,  509004,  214676,  54649,   8624,   826,   44,  1;
  11: 3628800, 6999840, 5753736, 2655764, 761166, 140889, 16884, 1266, 54, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n-1)*x)*P(n-1,x) for n >= 3, with P(2,x) = 1.
%t Mathematica
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n - 1]], {n, 2, 11}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 1)); v[1] = 1; for(n = 3, N, my(i = n - 1); v[i] = (1 + (n - 1)*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); 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(11)

Generated with polytool c48629374597.

A143491: reversed triangle, offset 2, from the row P0
%e Example
Triangle begins:
   2:       1;
   3:       2,       1;
   4:       6,       5,       1;
   5:      24,      26,       9,       1;
   6:     120,     154,      71,      14,      1;
   7:     720,    1044,     580,     155,     20,      1;
   8:    5040,    8028,    5104,    1665,    295,     27,     1;
   9:   40320,   69264,   48860,   18424,   4025,    511,    35,    1;
  10:  362880,  663696,  509004,  214676,  54649,   8624,   826,   44,  1;
  11: 3628800, 6999840, 5753736, 2655764, 761166, 140889, 16884, 1266, 54, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n-1)*x)*P(n-1,x) for n >= 3, with P(2,x) = 1.
%t Mathematica
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n - 1]], {n, 2, 11}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 1)); v[1] = 1; for(n = 3, N, my(i = n - 1); v[i] = (1 + (n - 1)*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); 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(11)

Generated with polytool c48629374597.

A145324: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  5,    6;
   4: 1,  9,   26,    24;
   5: 1, 14,   71,   154,    120;
   6: 1, 20,  155,   580,   1044,    720;
   7: 1, 27,  295,  1665,   5104,   8028,    5040;
   8: 1, 35,  511,  4025,  18424,  48860,   69264,   40320;
   9: 1, 44,  826,  8624,  54649, 214676,  509004,  663696,  362880;
  10: 1, 54, 1266, 16884, 140889, 761166, 2655764, 5753736, 6999840, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (1+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[(1 + 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] = (1 + 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.

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-4a8bd3a5d986eb7b
  • 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-07dfe744e3acc2e8, e-4000c0d63fa750fb, e-4a8bd3a5d986eb7b
  • 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-07dfe744e3acc2e8, e-4000c0d63fa750fb, e-4a8bd3a5d986eb7b
  • 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-4000c0d63fa750fb, e-4a8bd3a5d986eb7b
  • 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-4000c0d63fa750fb
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-07dfe744e3acc2e8

Proof chains

4 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 7 proof steps

Same fingerprint (2)

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