A269941 -id:A269941 - cp's OEIS frontend

A269942 Triangle read by rows, the coefficients of the inverse partial P-polynomials.

1, 0, -1, 0, -1, 1, 0, -2, 1, 2, -1, 0, -5, 5, -1, 5, -2, -3, 1, 0, -14, 21, -3, -6, 1, 14, -12, 2, -9, 3, 4, -1, 0, -42, 84, -28, -28, 7, 7, -1, 42, -56, 7, 14, -2, -28, 21, -3, 14, -4, -5, 1

Offset: 0

Peter Luschny, Mar 08 2016

The triangle of coefficients of the partial P-polynomials is A269941. For the definition of the inverse partial P-polynomials see the link 'P-transform'.

			[[1]],
[[0], [-1]],
[[0], [-1], [1]],
[[0], [-2, 1], [2], [-1]],
[[0], [-5, 5, -1], [5, -2], [-3], [1]],
[[0], [-14, 21, -3, -6, 1], [14, -12, 2], [-9, 3], [4], [-1]],
[[0], [-42,84,-28,-28,7,7,-1],[42,-56,7,14,-2],[-28,21,-3],[14,-4],[-5],[1]]
Replacing the sublists by their sums reduces the triangle to a signed version of the triangle A097805. The column 1 of sublists is A111785 in a different order.

Peter Luschny, The P-transform.

Cf. A097805, A111785, A268441, A268442, A269941.

# For function PMultiCoefficients see A269941.
PMultiCoefficients(7, inverse = True)

A356652 Triangle read by rows. Numerators of the coefficients of a sequence of rational polynomials r_n(x) with r_n(1) = B(2*n), where B(n) are the Bernoulli numbers.

Original entry on oeis.org

1, 0, 1, 0, 1, -1, 0, 1, -1, 5, 0, 1, -41, 14, -140, 0, 1, -23, 93, -40, 100, 0, 1, -157, 2948, -3652, 7700, -15400, 0, 1, -341, 18759, -1937936, 520520, -280280, 1401400, 0, 1, -1927, 3478, -7384676, 4364360, -1430000, 5605600, -8008000

Offset: 0

Peter Luschny, Sep 02 2022

			The rational triangle R(n, k) begins:
[0] 1;
[1] 0,     1/6;
[2] 0,    1/70,      -1/21;
[3] 0,   1/434,      -1/31,       5/93;
[4] 0,  1/2286,   -41/1905,     14/127,  -140/1143;
[5] 0, 1/11242,   -23/1533,     93/511,     -40/73,   100/219;
[6] 0, 1/53222, -157/14329, 2948/10235, -3652/2047, 7700/2047, -15400/6141;
.
Row sums are: 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, ... (A000367/A002445).

Peter Luschny, The P-transform.

Cf. A356653 (denominators), A269941, A000367, A002445.

# Using function PTrans from A269941.
R_row := n -> seq(coeffs(p), p in PTrans(n, n -> 1/((2*n)*(2*n + 1)),
n -> (2*n)!/(2-2^(2*n)))): seq(seq(numer(r), r in R_row(n)), n = 0..8);

Let r_n(x) = ((2*n)! / (2-2^(2*n))) * Sum_{p in P_n} (-x)^(p_1) * binomial(p_1, p_2) * binomial(p_2, p_3) * ... * binomial(p_{n-1}, p_{n}) * (2*3)^(-p_1) * (4*5)^(-p_2) * ... * (2*n*(2*n+1))^(-p_n), where P_n are the partitions of n and we say that p is a partition of n if and only if p = (p_{1}, ..., p_{n}), the p_{i} are integers, Sum_{1<=i<=n} p_i = n and p_{1} >= p_{2} >= ... >= p_{n} >= 0.

T(n, k) = numerator([x^k] r_n(x)).

A356653 Triangle read by rows. Denominators of the coefficients of a sequence of rational polynomials r_n(x) with r_n(1) = B(2*n), where B(n) are the Bernoulli numbers.

Original entry on oeis.org

1, 1, 6, 1, 70, 21, 1, 434, 31, 93, 1, 2286, 1905, 127, 1143, 1, 11242, 1533, 511, 73, 219, 1, 53222, 14329, 10235, 2047, 2047, 6141, 1, 245730, 40955, 40955, 368595, 24573, 8191, 73719, 1, 1114078, 294903, 4681, 491505, 42129, 4681, 14043, 42129

Offset: 0

Peter Luschny, Sep 02 2022

For formulas and comments see A356652.

			The triangle T(n, k) begins:
[0] 1;
[1] 1,     6;
[2] 1,    70,    21;
[3] 1,   434,    31,    93;
[4] 1,  2286,  1905,   127, 1143;
[5] 1, 11242,  1533,   511,   73,  219;
[6] 1, 53222, 14329, 10235, 2047, 2047, 6141;

Cf. A356652 (numerators), A269941.

# Using function PTrans from A269941.
R_row := n -> seq(coeffs(p), p in PTrans(n, n -> 1/((2*n)*(2*n + 1)),
n -> (2*n)!/(2-2^(2*n)))): seq(lprint(seq(denom(r), r in R_row(n))), n=0..9);

T(n, k) = denominator([x^k] r_n(x)), where the polynomials r_n(x) are defined in A356652.

A260533 Table of partition coefficients read by rows. The coefficient of a partition p is Product_{j=1..length(p)-1} C(p[j], p[j+1]). Row n lists the coefficients of the partitions of n in the ordering A080577, for n>=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 4, 3, 3, 2, 2, 1, 1, 5, 6, 4, 1, 6, 3, 1, 2, 2, 1, 1, 6, 10, 5, 4, 12, 4, 3, 3, 6, 3, 2, 2, 2, 1, 1, 7, 15, 6, 10, 20, 5, 1, 12, 6, 12, 4, 3, 3, 6, 6, 3, 1, 2, 2, 2, 1

Offset: 1

Peter Luschny, Jul 28 2015

The triangle is a refinement of Pascal's triangle A007318.

			The signed version of the triangle starts:
[1]
[-1, 1]
[1, -2, 1]
[-1, 3, -1, -2, 1]
[1, -4, 3, 3, -2, -2, 1]
[-1, 5, -6, -4, 1, 6, 3, -1, -2, -2, 1]
Adding adjacent coefficients with equal sign reduces the triangle to the matrix inverse of Pascal's triangle (A130595).
.
The q-polynomials defined by Cigler start:
[0]  1;
[1]  1, q;
[2]  1, 2*q, q^3;
[3]  1, 3*q, q^2+2*q^3,   q^6;
[4]  1, 4*q, 3*q^2+3*q^3, 2*q^4+2*q^6,     q^10;
[5]  1, 5*q, 6*q^2+4*q^3, q^3+6*q^4+3*q^6, q^6+2*q^7+2*q^10, q^15;

Johann Cigler, Some elementary remarks on the powers of a partial theta function and corresponding q-analogs of the binomial coefficients, arXiv:2408.14094 [math.NT], 2024.
Peter Luschny, The P-transform, 2016.
Peter Luschny, The Partition Transform, A SageMath Jupyter Notebook, GitHub, 2016/2022.
Marko Riedel, Answer to Question 4943578, Mathematics Stack Exchange, 2024.
Peter Taylor, Answer to Question 474483, MathOverflow, 2024.

Cf. A007318, A080577, A130595, A269941 (expanded form).

with(combstruct): with(ListTools):
PartitionCoefficients := proc(n) local L, iter, p;
iter := iterstructs(Partition(n)): L := []:
while not finished(iter) do
   p := Reverse(nextstruct(iter)):
   L := [mul(binomial(p[j], p[j+1]), j=1..nops(p)-1), op(L)]
od end:
for n from 1 to 6 do PartitionCoefficients(n) od;
# Alternative, using Cigler's recurrence for the q-polynomials:
C := proc(n, k, q) local j;
if k = 0 then q^binomial(n + 1, 2) elif n = 0 then n^k else
add(q^binomial(j + 1, 2)*C(n - j - 1, k - 1, q), j = 0..n - k) fi end:
p := n -> local k; add(C(n, n - k, q)*x^k, k = 0..n):
row := n -> local k; seq(sort(coeff(expand(p(n)), x, k), [q], ascending), k=0..n):
for n from 0 to 5 do row(n) od;  # Peter Luschny, Aug 24 2024

PartitionCoeff = lambda p: mul(binomial(p[j], p[j+1]) for j in range(len(p)-1))
PartitionCoefficients = lambda n: [PartitionCoeff(p) for p in Partitions(n)]
for n in (1..7): print(PartitionCoefficients(n))

Let P = Partitions(n, k) denote the set of partitions p of n with largest part k. Then Sum_{p in P} PartitionCoefficient(p) = binomial(n-1,k-1) for n>=0 and k>=0 (assuming binomial(-1,-1) = 1).

A269943 Triangle read by rows, T(n,k) = ((-1)^k(2n)!/4^k)P[n,k](1/((2n-1)(2n))) where P is the inverse P-transform, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 6, 0, 16, 60, 90, 0, 288, 1176, 2520, 2520, 0, 9216, 39360, 98280, 151200, 113400, 0, 460800, 2023296, 5504400, 10311840, 12474000, 7484400, 0, 33177600, 148442112, 426666240, 896575680, 1362160800, 1362160800, 681080400

Offset: 0

Peter Luschny, Mar 27 2016

The P-transform is defined in the link. Compare also the Sage implementation below.

			Triangle starts:
[1]
[0, 1]
[0, 2, 6]
[0, 16, 60, 90]
[0, 288, 1176, 2520, 2520]
[0, 9216, 39360, 98280, 151200, 113400]
[0, 460800, 2023296, 5504400, 10311840, 12474000, 7484400]

Peter Luschny, The P-transform.

Cf. A000680, A055546.

# uses[PtransMatrix from A269941]
eul = lambda n: 1/((2*n-1)*(2*n))
norm = lambda n,k: (-1)^k*factorial(2*n)/4^k
PtransMatrix(7, eul, norm, inverse=True)

T(n,1) = 2^(n-1)*(n-1)!^2 (cf. A055546) for n>=1.

T(n,n) = (2*n)!/2^n = A000680(n).

A356900 a(n) = P(n, 1/2) where P(n, x) = x^(-n)Sum_{k=0..n} A241171(n, k)x^k.

Original entry on oeis.org

1, 1, 8, 154, 5552, 321616, 27325088, 3200979664, 494474723072, 97390246272256, 23820397371219968, 7083386168647642624, 2516691244849530785792, 1052914814802404260765696, 512347915163742179541659648, 286902390859642414913802102784, 183187476890368376930869730803712

Offset: 0

Peter Luschny, Sep 03 2022

nonn

Other special values of this Euler type polynomials are: P(n, -1) = A000364(n); P(n, -1/2) = A002105(n); P(n, 1) = A094088(n), where we always make the assumption that the offset of the sequences is 0. A partition refinement of Joffe's triangle A241171 is A327022.

Cf. A241171, A327022, A000364, A002105, A094088, A269941.

a := n -> 2^n*add(A241171(n, k)*(1/2)^k, k = 0..n):
seq(a(n), n = 0..16);

# Using function PtransMatrix from A269941.
def E(n, v):
    eulr = lambda n: 1 / ((2 * n - 1) * (2 * n))
    norm = lambda n, k: (1 / v)^n * factorial(2 * n)
    P = PtransMatrix(n, eulr, norm)
    return [(-1)^j * sum([v^k * P[j][k] for k in range(j + 1)]) for j in range(n)]
A356900List = lambda n: E(n, -1/2); print(A356900List(17))
# A002105List = lambda n: E(n, 1/2) returns the reduced tangent numbers A002105.

cp's OEIS Frontend

A269942 Triangle read by rows, the coefficients of the inverse partial P-polynomials.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

A356652 Triangle read by rows. Numerators of the coefficients of a sequence of rational polynomials r_n(x) with r_n(1) = B(2*n), where B(n) are the Bernoulli numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A356653 Triangle read by rows. Denominators of the coefficients of a sequence of rational polynomials r_n(x) with r_n(1) = B(2*n), where B(n) are the Bernoulli numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A260533 Table of partition coefficients read by rows. The coefficient of a partition p is Product_{j=1..length(p)-1} C(p[j], p[j+1]). Row n lists the coefficients of the partitions of n in the ordering A080577, for n>=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Sage

Formula

A269943 Triangle read by rows, T(n,k) = ((-1)^k*(2*n)!/4^k)*P[n,k](1/((2*n-1)*(2*n))) where P is the inverse P-transform, for n>=0 and 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

Formula

A356900 a(n) = P(n, 1/2) where P(n, x) = x^(-n)*Sum_{k=0..n} A241171(n, k)*x^k.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

SageMath

A269943 Triangle read by rows, T(n,k) = ((-1)^k(2n)!/4^k)P[n,k](1/((2n-1)(2n))) where P is the inverse P-transform, for n>=0 and 0<=k<=n.

A356900 a(n) = P(n, 1/2) where P(n, x) = x^(-n)Sum_{k=0..n} A241171(n, k)x^k.