A373428
Triangle read by rows: Coefficients of the polynomials S2(n, x) * EZ(n, x), where S2 denote the Stirling set polynomials and EZ the Eulerian zig-zag polynomials A205497.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 1, 4, 4, 1, 0, 1, 10, 28, 26, 9, 1, 0, 1, 22, 137, 291, 261, 102, 17, 1, 0, 1, 45, 555, 2300, 4150, 3517, 1479, 306, 29, 1, 0, 1, 89, 2048, 15152, 48942, 76259, 61846, 26976, 6388, 795, 47, 1
Offset: 0
Tracing the computation:
0: [1] * [1] = [1]
1: [1] * [0, 1] = [0, 1]
2: [1] * [0, 1, 1] = [0, 1, 1]
3: [1, 1] * [0, 1, 3, 1] = [0, 1, 4, 4, 1]
4: [1, 3, 1] * [0, 1, 7, 6, 1] = [0, 1, 10, 28, 26, 9, 1]
5: [1, 7, 7, 1] * [0, 1, 15, 25, 10, 1] = [0, 1, 22, 137, 291, 261, 102, 17, 1]
A373426
Triangle read by rows: Coefficients of the polynomials L(n, x) * EZ(n, x), where L denote the unsigned Lah polynomials and EZ the Eulerian zig-zag polynomials A205497.
Original entry on oeis.org
1, 0, 1, 0, 2, 1, 0, 6, 12, 7, 1, 0, 24, 108, 144, 73, 15, 1, 0, 120, 1080, 2640, 2660, 1221, 267, 27, 1, 0, 720, 11880, 48720, 82980, 67350, 28321, 6344, 751, 44, 1, 0, 5040, 146160, 955080, 2529240, 3262350, 2245782, 870283, 195074, 25267, 1831, 68, 1
Offset: 0
Tracing the computation:
0: [1] * [1] = [1]
1: [1] * [0, 1] = [0, 1]
2: [1] * [0, 2, 1] = [0, 2, 1]
3: [1, 1] * [0, 6, 6, 1] = [0, 6, 12, 7, 1]
4: [1, 3, 1] * [0, 24, 36, 12, 1] = [0, 24, 108, 144, 73, 15, 1]
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# Using function EZP from A373432.
EZP((n, k) -> ifelse(n=k, 1, binomial(n-1, k-1)*n!/k!), 7);
A373429
Triangle read by rows: Coefficients of the polynomials S1(n, x) * EZ(n, x), where S1 denote the Stirling1 polynomials and EZ the Eulerian zig-zag polynomials A205497.
Original entry on oeis.org
1, 0, 1, 0, -1, 1, 0, 2, -1, -2, 1, 0, -6, -7, 21, -6, -3, 1, 0, 24, 118, -147, -91, 126, -28, -3, 1, 0, -120, -1406, -109, 3749, -2084, -450, 514, -94, -1, 1, 0, 720, 16956, 34240, -72307, -15475, 56286, -21125, -674, 1635, -262, 5, 1
Offset: 0
Tracing the computation:
0: [1] * [1] = [1]
1: [1] * [0, 1] = [0, 1]
2: [1] * [0, -1, 1] = [0, -1, 1]
3: [1, 1] * [0, 2, -3, 1] = [0, 2, -1, -2, 1]
4: [1, 3, 1] * [0, -6, 11, -6, 1] = [0, -6, -7, 21, -6, -3, 1]
5: [1, 7, 7, 1] * [0, 24, -50, 35, -10, 1] = [0, 24, 118, -147, -91, 126,-28,-3,1]
A373427
Triangle read by rows: Coefficients of the polynomials SC(n, x) * EZ(n, x), where SC denote the Stirling cycle polynomials and EZ the Eulerian zig-zag polynomials A205497.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 2, 5, 4, 1, 0, 6, 29, 45, 30, 9, 1, 0, 24, 218, 553, 629, 366, 112, 17, 1, 0, 120, 1954, 7781, 13409, 12136, 6270, 1894, 326, 29, 1, 0, 720, 20484, 125968, 313715, 407297, 308286, 143725, 42124, 7683, 830, 47, 1
Offset: 0
Tracing the computation:
0: [1] * [1] = [1]
1: [1] * [0, 1] = [0, 1]
2: [1] * [0, 1, 1] = [0, 1, 1]
3: [1, 1] * [0, 2, 3, 1] = [0, 2, 5, 4, 1]
4: [1, 3, 1] * [0, 6, 11, 6, 1] = [0, 6, 29, 45, 30, 9, 1]
5: [1, 7, 7, 1] * [0, 24, 50, 35, 10, 1] = [0, 24, 218, 553, 629, 366, 112,17,1]
A373572
Triangle read by rows: Coefficients of the polynomials P(n, x) * EZ(n, x), where P denote the signed Pascal polynomials and EZ the Eulerian zig-zag polynomials A205497.
Original entry on oeis.org
1, -1, 1, 1, -2, 1, -1, 2, 0, -2, 1, 1, -1, -5, 10, -5, -1, 1, -1, -2, 18, -26, 0, 26, -18, 2, 1, 1, 8, -38, 18, 117, -212, 117, 18, -38, 8, 1, -1, -19, 52, 143, -677, 818, 0, -818, 677, -143, -52, 19, 1, 1, 38, -6, -817, 2196, -722, -5071, 8762, -5071, -722, 2196, -817, -6, 38, 1
Offset: 0
Triangle starts:
[0] [1]
[1] [-1, 1]
[2] [ 1, -2, 1]
[3] [-1, 2, 0, -2, 1]
[4] [ 1, -1, -5, 10, -5, -1, 1]
[5] [-1, -2, 18, -26, 0, 26, -18, 2, 1]
[6] [ 1, 8, -38, 18, 117, -212, 117, 18, -38, 8, 1]
[7] [-1, -19, 52, 143, -677, 818, 0, -818, 677, -143, -52, 19, 1]
Showing 1-5 of 5 results.
