A159854 → D

OEIS matches ?

• A045623: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A128710: triangle, 68 terms agree, 11 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A196410: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 5 ?

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, 2;
  1: 0, 2,  3;
  2: 0, 2,  6,   4;
  3: 0, 2,  9,  12,   5;
  4: 0, 2, 12,  24,  20,   6;
  5: 0, 2, 15,  40,  50,  30,   7;
  6: 0, 2, 18,  60, 100,  90,  42,   8;
  7: 0, 2, 21,  84, 175, 210, 147,  56,   9;
  8: 0, 2, 24, 112, 280, 420, 392, 224,  72, 10;
  9: 0, 2, 27, 144, 420, 756, 882, 672, 324, 90, 11;

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

P[0, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (n + 4)*x + (n + 2)*x^2)*P[n - 1, x])/(2 + (n + 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (n + 4)*x + (n + 2)*x^2)*v[i - 1])/(2 + (n + 1)*x)); 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: 2;
  1: 2,  3;
  2: 2,  6,   4;
  3: 2,  9,  12,   5;
  4: 2, 12,  24,  20,   6;
  5: 2, 15,  40,  50,  30,   7;
  6: 2, 18,  60, 100,  90,  42,   8;
  7: 2, 21,  84, 175, 210, 147,  56,   9;
  8: 2, 24, 112, 280, 420, 392, 224,  72, 10;
  9: 2, 27, 144, 420, 756, 882, 672, 324, 90, 11;

Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where (2+(n+1)*x)*P(n,x) = (2+(n+4)*x+(n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2*x.

P[0, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (n + 4)*x + (n + 2)*x^2)*P[n - 1, x])/(2 + (n + 1)*x)]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (n + 4)*x + (n + 2)*x^2)*v[i - 1])/(2 + (n + 1)*x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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-23d1d4324cad1a70
• 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-23d1d4324cad1a70
• 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-23d1d4324cad1a70

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-23d1d4324cad1a70

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

  • premise (sequence): Riordan array (1-x,x/(1-x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-b62074ba45057e82 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

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

• A007318 → (1/2+(1/2)n)θ+1+n → reverse
• A124932: Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).
• A007318 → θ+1 → D

References

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