Unsigned 2-Stirling numbers of the first kind.

OEIS matches ?

• A188588: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A357711: 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:         6,          5,         1;
  1:        24,         26,         9,         1;
  2:       120,        154,        71,        14,         1;
  3:       720,       1044,       580,       155,        20,        1;
  4:      5040,       8028,      5104,      1665,       295,       27,       1;
  5:     40320,      69264,     48860,     18424,      4025,      511,      35,      1;
  6:    362880,     663696,    509004,    214676,     54649,     8624,     826,     44,     1;
  7:   3628800,    6999840,   5753736,   2655764,    761166,   140889,   16884,   1266,    54,    1;
  8:  39916800,   80627040,  70290936,  34967140,  11028590,  2310945,  326613,  30810,  1860,   65,  1;
  9: 479001600, 1007441280, 924118272, 489896616, 167310220, 38759930, 6230301, 696333, 53130, 2640, 77, 1;

The row polynomials satisfy P(n,x) = (n+3+x)*P(n-1,x) for n >= 1, with P(0,x) = 6 + 5*x + x^2.

P[0, x_] := 6 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n + 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] = 6 + 5*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 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-84f3431e657be6ed
• 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-25f97869157cba20, e-84f3431e657be6ed
• 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-25f97869157cba20, e-84f3431e657be6ed
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-84f3431e657be6ed
• 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-25f97869157cba20
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, 0]: (-inf, 0] -> (-inf, 0] through R_s, s = 1, and U_g
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-aadeee73ad838eee

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map_theorem e-aadeee73ad838eee root: theorem check

  Index-dependent basis map [Bre89, Thm 2.4.3; MW08, Thm 2] polytool RealRooted basisMap_factorization_of_scalarIndex_recurrence RealRooted rayBasisMap_maps_realRootedIn_iff ?

• parent_provers root_domain e-25f97869157cba20 root: theorem check

  Parent-row provers of polytool polytool ?

• theorem_check liu_wang e-84f3431e657be6ed root: theorem check

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

Used in 7 proof steps

• e-120de3004347d127 basis_map (A013611) → basis_map_A013611 of Unsigned 2-Stirling numbers of the first kind.
• e-13b78624414ee2b1 basis_map (A008277) → basis_map_A008277 of Unsigned 2-Stirling numbers of the first kind.
• e-36015c56c1237184 basis_map (A013610) → basis_map_A013610 of Unsigned 2-Stirling numbers of the first kind.
• e-7cc986a9eefcf6a5 basis_map (A013609) → basis_map_A013609 of Unsigned 2-Stirling numbers of the first kind.
• e-a33c0ab38a356260 basis_map (A007318) → basis_map_A007318 of Unsigned 2-Stirling numbers of the first kind.
• e-a410505258437c48 obreschkoff_combination → obreschkoff_combination of Unsigned 2-Stirling numbers of the first kind.
• e-a410505258437c48 obreschkoff_combination → obreschkoff_combination of Unsigned 2-Stirling numbers of the first kind.

Related sequences ?

Same fingerprint (2)

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

• A136124: Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers o…
• A145324: Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....

References

1. [Bre89] Francesco Brenti. Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc., 81(413):viii+106, 1989. doi:10.1090/memo/0413
2. [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
3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741