A101280: Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).

OEIS matches ?

• A101280: triangle, 46 terms agree, 12 full rows row lengths from the atlas (tabf) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 992 terms from n = 1 ?
• A080635: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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;
  2: 1,    2;
  3: 1,    8;
  4: 1,   22,    16;
  5: 1,   52,   136;
  6: 1,  114,   720,    272;
  7: 1,  240,  3072,   3968;
  8: 1,  494, 11616,  34304,   7936;
  9: 1, 1004, 40776, 230144, 176896;

The row polynomials satisfy P(n,x) = (1+(2*n-2)*x)*P(n-1,x) + (x-4*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[(1 + (2*n - 2)*x)*P[n - 1, x] + (x - 4*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] = (1 + (2*n - 2)*x)*v[i - 1] + (x - 4*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:
   1: 1;
   2: 1;
   3: 1,    2;
   4: 1,    8;
   5: 1,   22,    16;
   6: 1,   52,   136;
   7: 1,  114,   720,    272;
   8: 1,  240,  3072,   3968;
   9: 1,  494, 11616,  34304,   7936;
  10: 1, 1004, 40776, 230144, 176896;

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

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n - 4)*x)*P[n - 1, x] + (x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (2*n - 4)*x)*v[i - 1] + (x - 4*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, 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-e49ea542bfed6bbf
• 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-e49ea542bfed6bbf
• 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-e49ea542bfed6bbf

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

• gamma_polynomial scale 1, shift 1 e-e49ea542bfed6bbf

  Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

  • premise (sequence): eulerian
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-ab3cdcfae289b48d root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers root_domain e-6c46c948823f0a8c root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image reversal, shift 0 e-c61ecf4ef1c45372

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): eulerian
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-ab3cdcfae289b48d (expanded above)
        • ↵ parent_provers root_domain e-6c46c948823f0a8c (expanded above)

Used in 5 proof steps

• e-019ad329539bbbb3 preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows…
• e-2394f256e051d5b6 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, w…
• e-33f9ec87155c6d6e preserver_image (affine, shift 0) → shift_x_plus_one of Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, wher…
• e-34f81fb2d053d311 preserver_image (derivative_polynomial, shift 0) → derivative of Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,…
• e-f727bc245399bdee preserver_image (euler_polynomial, shift 0) → theta_plus_one of Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where …

Related sequences ?

Same fingerprint (3)

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

• A008292 → γ
• A257620 → γ
• A257609 → γ

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003