A152842 → reverse

OEIS matches ?

• A094015: 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 ?
• A152842: reversed triangle, 63 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 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:   1;
  1:   3,    1;
  2:   3,    4,    1;
  3:   9,   15,    7,    1;
  4:   9,   24,   22,    8,    1;
  5:  27,   81,   90,   46,   11,    1;
  6:  27,  108,  171,  136,   57,   12,   1;
  7:  81,  351,  621,  579,  307,   93,  15,   1;
  8:  81,  432,  972, 1200,  886,  400, 108,  16,  1;
  9: 243, 1377, 3348, 4572, 3858, 2086, 724, 156, 19, 1;

The row polynomials satisfy P(n,x) = -(4*n-7+(2*n-3)*x)*P(n-1,x) + (6+8*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4*n - 7 + (2*n - 3)*x)*P[n - 1, x] + (6 + 8*x + 2*x^2)*D[P[n - 1, x], 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] = -(4*n - 7 + (2*n - 3)*x)*v[i - 1] + (6 + 8*x + 2*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 - 1, v[i])))};
T(9)

Triangle begins:
  0: 1;
  1: 1,  3;
  2: 1,  4,   3;
  3: 1,  7,  15,   9;
  4: 1,  8,  22,  24,    9;
  5: 1, 11,  46,  90,   81,   27;
  6: 1, 12,  57, 136,  171,  108,   27;
  7: 1, 15,  93, 307,  579,  621,  351,   81;
  8: 1, 16, 108, 400,  886, 1200,  972,  432,   81;
  9: 1, 19, 156, 724, 2086, 3858, 4572, 3348, 1377, 243;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = -(4*n-7+(2*n-3)*x)*P(n-1,x) + (6+8*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4*n - 7 + (2*n - 3)*x)*P[n - 1, x] + (6 + 8*x + 2*x^2)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(4*n - 7 + (2*n - 3)*x)*v[i - 1] + (6 + 8*x + 2*x^2)*deriv(v[i - 1], x)); 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(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-67f427bf8180eb6f
• 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-67f427bf8180eb6f
• 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-67f427bf8180eb6f

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 reversal, shift 0 e-67f427bf8180eb6f

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-596b986e3a00db44 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.

• A152842: Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] …