A144084: T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.

OEIS matches ?

• A002720: 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 ?
• A144084: triangle, 45 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,  9,   18,     6;
  4: 1, 16,   72,    96,     24;
  5: 1, 25,  200,   600,    600,     120;
  6: 1, 36,  450,  2400,   5400,    4320,     720;
  7: 1, 49,  882,  7350,  29400,   52920,   35280,    5040;
  8: 1, 64, 1568, 18816, 117600,  376320,  564480,  322560,   40320;
  9: 1, 81, 2592, 42336, 381024, 1905120, 5080320, 6531840, 3265920, 362880;

The row polynomials satisfy P(n,x) = (1+(2*n-1)*x)*P(n-1,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 + (2*n - 1)*x)*P[n - 1, 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 + (2*n - 1)*x)*v[i - 1] - 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,  9,   18,     6;
  4: 1, 16,   72,    96,     24;
  5: 1, 25,  200,   600,    600,     120;
  6: 1, 36,  450,  2400,   5400,    4320,     720;
  7: 1, 49,  882,  7350,  29400,   52920,   35280,    5040;
  8: 1, 64, 1568, 18816, 117600,  376320,  564480,  322560,   40320;
  9: 1, 81, 2592, 42336, 381024, 1905120, 5080320, 6531840, 3265920, 362880;

The row polynomials satisfy P(n,x) = (1+(2*n-1)*x)*P(n-1,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 + (2*n - 1)*x)*P[n - 1, 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 + (2*n - 1)*x)*v[i - 1] - 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 (-inf, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-c98efb5092e3dc49
• 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-c98efb5092e3dc49
• 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-c98efb5092e3dc49
• 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-c98efb5092e3dc49

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

• theorem_check liu_wang e-c98efb5092e3dc49 root: theorem check

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

Used in 3 proof steps

• e-50ecd620d262c41e obreschkoff_combination → obreschkoff_combination of T(n,k) is the number of partial bijections of height k (height…
• e-50ecd620d262c41e obreschkoff_combination → obreschkoff_combination of T(n,k) is the number of partial bijections of height k (height…
• e-e31dcd4b09a7ff12 preserver_image (affine, shift -1) → Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).

Related sequences ?

Same fingerprint (2)

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

• Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).
• A132393 → basis A008277

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