A039755: Triangle of B-analogs of Stirling numbers of the second kind.

OEIS matches ?

• A039756: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A007405: 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 ?
• A039755: triangle, 56 terms agree, 10 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:
   1: 1;
   2: 1,    1;
   3: 1,    4,      1;
   4: 1,   13,      9,      1;
   5: 1,   40,     58,     16,      1;
   6: 1,  121,    330,    170,     25,      1;
   7: 1,  364,   1771,   1520,    395,     36,     1;
   8: 1, 1093,   9219,  12411,   5075,    791,    49,    1;
   9: 1, 3280,  47188,  96096,  58086,  13776,  1428,   64,  1;
  10: 1, 9841, 239220, 719860, 618870, 209622, 32340, 2388, 81, 1;

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

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

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + 2*x*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, v[i])))};
T(10)

Triangle begins:
  0: 1;
  1: 1,    1;
  2: 1,    4,      1;
  3: 1,   13,      9,      1;
  4: 1,   40,     58,     16,      1;
  5: 1,  121,    330,    170,     25,      1;
  6: 1,  364,   1771,   1520,    395,     36,     1;
  7: 1, 1093,   9219,  12411,   5075,    791,    49,    1;
  8: 1, 3280,  47188,  96096,  58086,  13776,  1428,   64,  1;
  9: 1, 9841, 239220, 719860, 618870, 209622, 32340, 2388, 81, 1;

The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 2*x*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*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*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)

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,    1;
  3: 1,  9,   13,     1;
  4: 1, 16,   58,    40,      1;
  5: 1, 25,  170,   330,    121,      1;
  6: 1, 36,  395,  1520,   1771,    364,      1;
  7: 1, 49,  791,  5075,  12411,   9219,   1093,      1;
  8: 1, 64, 1428, 13776,  58086,  96096,  47188,   3280,    1;
  9: 1, 81, 2388, 32340, 209622, 618870, 719860, 239220, 9841, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + 2*x*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*D[P[n - 1, x], x]];
Table[Reverse[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*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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 >= 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-c873e8cf4f19d199
• 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-c873e8cf4f19d199
• 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-c873e8cf4f19d199
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-c873e8cf4f19d199
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 1] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 1] with m = 0, target (-inf, 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-fa5587de5743fdee

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-fa5587de5743fdee root: theorem check

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

• theorem_check liu_wang e-c873e8cf4f19d199 root: theorem check

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

Used in 9 proof steps

• e-a691d30fc1b4b246 obreschkoff_combination → obreschkoff_combination of Triangle of B-analogs of Stirling numbers of the second kind.
• e-a691d30fc1b4b246 obreschkoff_combination → obreschkoff_combination of Triangle of B-analogs of Stirling numbers of the second kind.
• e-25255a5163ba8ab9 preserver_image (affine, shift 0) → shift_x_plus_one of Triangle of B-analogs of Stirling numbers of the second kind.
• e-4980121166fd7e9f preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle of B-analogs of Stirling numbers of the second kind.
• e-694ead71eec92686 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Triangle of B-analogs of Stirling numbers of the second kind.
• e-70d96bc14ba32af8 preserver_image (reversal, shift 0) → reversal of Triangle of B-analogs of Stirling numbers of the second kind.
• e-a5b6b9e544860989 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Triangle of B-analogs of Stirling numbers of the second kind.
• e-c92588563eed0abe preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Triangle of B-analogs of Stirling numbers of the second kind.
• e-e4e3f503de69bba7 preserver_image (derivative_polynomial, shift 0) → derivative of Triangle of B-analogs of Stirling numbers of the second kind.

Related sequences ?

Same fingerprint (3)

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

• A039756: Triangle of B-analogs of Stirling numbers of 2nd kind.
• A039756 → reverse
• A039755 → reverse

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
2. [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