A008292 → x ↦ x+1

OEIS matches ?

• A000670: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A034172: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A090582: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
• A019538: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

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.

Triangle begins:
   1:       1;
   2:       2,        1;
   3:       6,        6,        1;
   4:      24,       36,       14,        1;
   5:     120,      240,      150,       30,        1;
   6:     720,     1800,     1560,      540,       62,       1;
   7:    5040,    15120,    16800,     8400,     1806,     126,      1;
   8:   40320,   141120,   191520,   126000,    40824,    5796,    254,     1;
   9:  362880,  1451520,  2328480,  1905120,   834120,  186480,  18150,   510,    1;
  10: 3628800, 16329600, 30240000, 29635200, 16435440, 5103000, 818520, 55980, 1022, 1;

The row polynomials satisfy P(n,x) = (n+(n-1)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + (n - 1)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (n + (n - 1)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Triangle begins:
   1: 1;
   2: 1,    2;
   3: 1,    6,     6;
   4: 1,   14,    36,     24;
   5: 1,   30,   150,    240,     120;
   6: 1,   62,   540,   1560,    1800,      720;
   7: 1,  126,  1806,   8400,   16800,    15120,     5040;
   8: 1,  254,  5796,  40824,  126000,   191520,   141120,    40320;
   9: 1,  510, 18150, 186480,  834120,  1905120,  2328480,  1451520,   362880;
  10: 1, 1022, 55980, 818520, 5103000, 16435440, 29635200, 30240000, 16329600, 3628800;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n+(n-1)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + (n - 1)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (n + (n - 1)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); 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)

Triangle begins:
   1:       1;
   2:       2,        1;
   3:       6,        6,        1;
   4:      24,       36,       14,        1;
   5:     120,      240,      150,       30,        1;
   6:     720,     1800,     1560,      540,       62,       1;
   7:    5040,    15120,    16800,     8400,     1806,     126,      1;
   8:   40320,   141120,   191520,   126000,    40824,    5796,    254,     1;
   9:  362880,  1451520,  2328480,  1905120,   834120,  186480,  18150,   510,    1;
  10: 3628800, 16329600, 30240000, 29635200, 16435440, 5103000, 818520, 55980, 1022, 1;

The row polynomials satisfy P(n,x) = (n+(n-1)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + (n - 1)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (n + (n - 1)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); 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)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-16a5ff84df4996fb
• Coefficients of one sign. Every nonzero row, n >= 1: 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-16a5ff84df4996fb
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-16a5ff84df4996fb
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-16a5ff84df4996fb

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

• preserver_image affine, shift 0 e-16a5ff84df4996fb

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): eulerian
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-ab3cdcfae289b48d root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers root_domain e-6c46c948823f0a8c root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image reversal, shift 0 e-c61ecf4ef1c45372

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): eulerian
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-ab3cdcfae289b48d (expanded above)
        • ↵ parent_provers root_domain e-6c46c948823f0a8c (expanded above)

Related sequences ?

Same fingerprint (5)

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

• A173018 → reverse (shift +1) → x ↦ x/(x+1)
• A019538: Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
• A131689 → reverse
• A131689: Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.
• A090582: T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1…

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