A269944 → basis A007318

OEIS matches ?

• A375041: reversed triangle, 46 terms agree, 8 full rows row lengths from the atlas (tabf) OEIS nonn · view at the OEIS · generated b-file 989 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:
  0:          1;
  1:          1,           1;
  2:          2,           3,          1;
  3:         10,          17,          8,          1;
  4:        100,         180,         97,         18,         1;
  5:       1700,        3160,       1829,        403,        35,        1;
  6:      44200,       83860,      50714,      12307,      1313,       61,      1;
  7:    1635400,     3147020,    1960278,     506073,     60888,     3570,     98,     1;
  8:   81770000,   158986400,  101160920,   27263928,   3550473,   239388,   8470,   148,   1;
  9: 5315050000, 10415886000, 6734446200, 1873316240, 258044673, 19110693, 789938, 18090, 213, 1;

The row polynomials satisfy P(n,x) = (n^2-2*n+2+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n^2 - 2*n + 2 + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

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

Triangle begins:
   1: 1,   1;
   2: 1,   3,     2;
   3: 1,   8,    17,      10;
   4: 1,  18,    97,     180,      100;
   5: 1,  35,   403,    1829,     3160,       1700;
   6: 1,  61,  1313,   12307,    50714,      83860,       44200;
   7: 1,  98,  3570,   60888,   506073,    1960278,     3147020,      1635400;
   8: 1, 148,  8470,  239388,  3550473,   27263928,   101160920,    158986400,     81770000;
   9: 1, 213, 18090,  789938, 19110693,  258044673,  1873316240,   6734446200,  10415886000,   5315050000;
  10: 1, 295, 35556, 2273318, 83885609, 1825121499, 23032979426, 160346377880, 562640474400, 859417702000, 435834100000;

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

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

P(N) = {my(v = vector(N)); v[1] = 1 + x; for(n = 2, N, my(i = n); v[i] = (n^2 - 2*n + 2 + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = Vecrev(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 >= 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-fc45cb3590dab8d3
• 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-fc45cb3590dab8d3
• 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-fc45cb3590dab8d3

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

• basis_map A007318 e-fc45cb3590dab8d3

  Geometric basis map (elementary) polytool ?

  • premise (sequence): 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)…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers root_domain e-aa0ec4b5da096d18 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.