A064861 → reverse

OEIS matches ?

• A026532: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:  2,   3,    1;
  1:  4,   8,    5,    1;
  2:  4,  12,   13,    6,    1;
  3:  8,  28,   38,   25,    8,    1;
  4:  8,  36,   66,   63,   33,    9,    1;
  5: 16,  80,  168,  192,  129,   51,   11,    1;
  6: 16,  96,  248,  360,  321,  180,   62,   12,   1;
  7: 32, 208,  592,  968, 1002,  681,  304,   86,  14,   1;
  8: 32, 240,  800, 1560, 1970, 1683,  985,  390, 100,  15,  1;
  9: 64, 512, 1840, 3920, 5500, 5336, 3653, 1765, 590, 130, 17, 1;

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

P[0, x_] := 2 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[-(3*n + 1 + (2*n + 1)*x)*P[n - 1, x] + (4 + 6*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] = 2 + 3*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = -(3*n + 1 + (2*n + 1)*x)*v[i - 1] + (4 + 6*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)

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-75e40465d2024023
• 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-75e40465d2024023
• 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-75e40465d2024023

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-75e40465d2024023

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-bbffc3f85cdc52e2 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

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

• A103633 → basis A013609
• Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.