A132393 → D

OEIS matches ?

• A028421: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A000254: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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:     0;
  1:     1;
  2:     1,      2;
  3:     2,      6,      3;
  4:     6,     22,     18,      4;
  5:    24,    100,    105,     40,      5;
  6:   120,    548,    675,    340,     75,     6;
  7:   720,   3528,   4872,   2940,    875,   126,    7;
  8:  5040,  26136,  39396,  27076,   9800,  1932,  196,   8;
  9: 40320, 219168, 354372, 269136, 112245, 27216, 3822, 288, 9;

The row polynomials satisfy P(n,x) = -(n^2-4*n+4+(2*n-4)*x+x^2)*P(n-2,x) + (2*n-3+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 4 + (2*n - 4)*x + x^2)*P[n - 2, x] + (2*n - 3 + 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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 4 + (2*n - 4)*x + x^2)*v[i - 2] + (2*n - 3 + 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:
  0:      1;
  1:      1,       2;
  2:      2,       6,       3;
  3:      6,      22,      18,       4;
  4:     24,     100,     105,      40,       5;
  5:    120,     548,     675,     340,      75,      6;
  6:    720,    3528,    4872,    2940,     875,    126,     7;
  7:   5040,   26136,   39396,   27076,    9800,   1932,   196,    8;
  8:  40320,  219168,  354372,  269136,  112245,  27216,  3822,  288,   9;
  9: 362880, 2053152, 3518100, 2894720, 1346625, 379638, 66150, 6960, 405, 10;

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

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

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

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-55ee0b1ee1acb060
• 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-55ee0b1ee1acb060
• 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-55ee0b1ee1acb060

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 derivative_polynomial, shift 0 e-55ee0b1ee1acb060

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers root_domain e-edbf92ee2e446027 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

• A130534 → θ+1

References

1. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.