A225476: Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

OEIS matches ?

• A337026: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A225476: triangle, 49 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

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,    1;
  2: 1,    4,      2;
  3: 1,   13,     18,       6;
  4: 1,   40,    116,      96,       24;
  5: 1,  121,    660,    1020,      600,      120;
  6: 1,  364,   3542,    9120,     9480,     4320,      720;
  7: 1, 1093,  18438,   74466,   121800,    94920,    35280,     5040;
  8: 1, 3280,  94376,  576576,  1394064,  1653120,  1028160,   322560,   40320;
  9: 1, 9841, 478440, 4319160, 14852880, 25154640, 23284800, 12035520, 3265920, 362880;

The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (2*x+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 + x)*P[n - 1, x] + (2*x + 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 + x)*v[i - 1] + (2*x + 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:
  0: 1;
  1: 1,    1;
  2: 1,    4,      2;
  3: 1,   13,     18,       6;
  4: 1,   40,    116,      96,       24;
  5: 1,  121,    660,    1020,      600,      120;
  6: 1,  364,   3542,    9120,     9480,     4320,      720;
  7: 1, 1093,  18438,   74466,   121800,    94920,    35280,     5040;
  8: 1, 3280,  94376,  576576,  1394064,  1653120,  1028160,   322560,   40320;
  9: 1, 9841, 478440, 4319160, 14852880, 25154640, 23284800, 12035520, 3265920, 362880;

The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (2*x+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 + x)*P[n - 1, x] + (2*x + x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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 + x)*v[i - 1] + (2*x + x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 (-2, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-0759b614e8dc4111
• 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-0759b614e8dc4111
• 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-0759b614e8dc4111
• 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-0759b614e8dc4111
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in [-2, 0]: source [-1, 1], target [-2, 0]
  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-1a33bcef7c8475f7

Proof chains

2 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-1a33bcef7c8475f7 root: theorem check

  Interval basis map theorem [MW08, Thm 2, Cor 3] polytool RealRooted intervalBasisMap_maps_realRootedIn_iff ?

• theorem_check ma_wang_interval e-0759b614e8dc4111 root: theorem check

  Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 2 proof steps

• e-b0c8196dbc01af81 obreschkoff_combination → obreschkoff_combination of Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are th…
• e-b0c8196dbc01af81 obreschkoff_combination → obreschkoff_combination of Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are th…

Related sequences ?

No candidate equalities or related sequences.

References

1. [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