A143491 → basis A013610

OEIS matches ?

• A001725: 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:         12,          21,           9;
  1:         60,         141,         108,          27;
  2:        360,        1026,        1071,         486,          81;
  3:       2520,        8262,       10575,        6615,        2025,         243;
  4:      20160,       73656,      109386,       84645,       36045,        8019,        729;
  5:     181440,      723384,     1205442,     1089963,      578340,      180306,      30618,       2187;
  6:    1814400,     7778160,    14224572,    14515956,     9053289,     3538080,     847098,     113724,      6561;
  7:   19958400,    91002960,   179804772,   202349232,   143134047,    66078747,   19932318,    3792258,    413343,    19683;
  8:  239500800,  1151910720,  2430666144,  2967605100,  2324656260,  1222347105,  437424057,  105304050,  16336890,  1476225,   59049;
  9: 3113510400, 15693341760, 35054392032, 45870864732, 39123346680, 22864481145, 9353554056, 2681224821, 528291720, 68201595, 5196312, 177147;

The row polynomials satisfy P(n,x) = (n+4+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 21*x + 9*x^2.

P[0, x_] := 12 + 21*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 4 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 21*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 4 + 3*x)*v[i - 1]); 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-36015c56c1237184
• 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-36015c56c1237184
• 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-36015c56c1237184

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 A013610 e-36015c56c1237184

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Unsigned 2-Stirling numbers of the first kind.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-84f3431e657be6ed root: theorem check

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

    • parent_provers root_domain e-25f97869157cba20 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.

• A136124 → basis A013610

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