Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.

OEIS matches ?

No OEIS matches.

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:
   1: 1,    7,       1;
   2: 1,   21,      21,        1;
   3: 1,   51,     161,       51,         1;
   4: 1,  113,     813,      813,       113,         1;
   5: 1,  239,    3361,     7631,      3361,       239,         1;
   6: 1,  493,   12421,    53833,     53833,     12421,       493,         1;
   7: 1, 1003,   42865,   320107,    607009,    320107,     42865,      1003,        1;
   8: 1, 2025,  141549,  1704693,   5494017,   5494017,   1704693,    141549,     2025,       1;
   9: 1, 4071,  453905,  8422679,  42924113,  72605303,  42924113,   8422679,   453905,    4071,    1;
  10: 1, 8165, 1426725, 39474449, 302458313, 802028813, 802028813, 302458313, 39474449, 1426725, 8165, 1;

The row polynomials satisfy P(n,x) = (1+(n+1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) + (n+2)*x*P(n-2,x) for n >= 3, with P(1,x) = 1 + 7*x + x^2, P(2,x) = 1 + 21*x + 21*x^2 + x^3.

P[1, x_] := 1 + 7*x + x^2;
P[2, x_] := 1 + 21*x + 21*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(1 + (n + 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] + (n + 2)*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 7*x + x^2; v[2] = 1 + 21*x + 21*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = (1 + (n + 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) + (n + 2)*x*v[i - 2]); 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 >= 1, 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-460746d5de499dd7
• Coefficients of one sign. Every nonzero row, n >= 1: 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-460746d5de499dd7
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-460746d5de499dd7
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-460746d5de499dd7

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

• theorem_check liu_wang_derivative_lag e-460746d5de499dd7 root: theorem check

  Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Used in 1 proof step

• e-ecadc04944221db5 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle read by rows, related to number of permutations of [n] with …

Related sequences ?

Same fingerprint (1)

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

• A271697: Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 a…

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