A133214 → basis A007318

OEIS matches ?

• A084771: 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:       13,        16,         4;
  1:       63,       114,        60,          8;
  2:      321,       768,       624,        192,         16;
  3:     1683,      5010,      5520,       2720,        560,        32;
  4:     8989,     32016,     44580,      30400,      10320,      1536,        64;
  5:    48639,    201698,    339612,     296520,     141680,     35616,      4032,       128;
  6:   265729,   1257472,   2484160,    2646784,    1637440,    587776,    114688,     10240,      256;
  7:  1462563,   7777314,  17635392,   22194816,   16890048,   7926912,   2236416,    350208,    25344,     512;
  8:  8097453,  47800080, 122356260,  177672960,  160668480,  93413376,  34755840,   7956480,  1025280,   61440,   1024;
  9: 45046719, 292292946, 833645340, 1372173000, 1438388160, 999132288, 463448832, 140997120, 26822400, 2900480, 146432, 2048;

The row polynomials satisfy (n+2)*P(n,x) = -(n+1+(4*n+4)*x+(4*n+4)*x^2)*P(n-2,x) + (6*n+9+(4*n+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 13 + 16*x + 4*x^2, P(1,x) = 63 + 114*x + 60*x^2 + 8*x^3.

P[0, x_] := 13 + 16*x + 4*x^2;
P[1, x_] := 63 + 114*x + 60*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (4*n + 4)*x + (4*n + 4)*x^2)*P[n - 2, x] + (6*n + 9 + (4*n + 6)*x)*P[n - 1, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 13 + 16*x + 4*x^2; v[2] = 63 + 114*x + 60*x^2 + 8*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + (4*n + 4)*x + (4*n + 4)*x^2)*v[i - 2] + (6*n + 9 + (4*n + 6)*x)*v[i - 1])/(n + 2)); 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-9a0dabc7f010818b
• 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-9a0dabc7f010818b
• 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-9a0dabc7f010818b
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-9a0dabc7f010818b

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

• basis_map A007318 e-9a0dabc7f010818b

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Delannoy paths counted by number of weak peaks.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-4ab8ec5aa6b36e54 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (1)

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

• A104684 → basis A013609

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023