A290311 -id:A290311 - cp's OEIS frontend

A290312 Third diagonal sequence of the Sheffer triangle A094816 (special Charlier).

1, 8, 29, 75, 160, 301, 518, 834, 1275, 1870, 2651, 3653, 4914, 6475, 8380, 10676, 13413, 16644, 20425, 24815, 29876, 35673, 42274, 49750, 58175, 67626, 78183, 89929, 102950, 117335, 133176, 150568, 169609, 190400, 213045, 237651, 264328, 293189, 324350, 357930

Offset: 0

Wolfdieter Lang, Jul 28 2017

See A094816 and A290311.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A094816, A290311.

Vec((1 + 3*x - x^2)/(1 - x)^5 + O(x^60)) \\ Colin Barker, Jul 29 2017

G.f.: (1 + 3*x - x^2)/(1 - x)^5.
E.g.f.: exp(x)*(1 + 7*x + 14*x^2/2! + 11*x^3/3! + 3*x^4/4!). This is computed from the o.g.f. with eqs. (23)-(25) of the Wolfdieter Lang 2017 link in A282629.
From Colin Barker, Jul 29 2017: (Start)
a(n) = (24 + 70*n + 69*n^2 + 26*n^3 + 3*n^4) / 24.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
(End)

A290313 Fourth diagonal sequence of the Sheffer triangle A094816 (special Charlier).

Original entry on oeis.org

1, 24, 145, 545, 1575, 3836, 8274, 16290, 29865, 51700, 85371, 135499, 207935, 309960, 450500, 640356, 892449, 1222080, 1647205, 2188725, 2870791, 3721124, 4771350, 6057350, 7619625, 9503676, 11760399, 14446495, 17624895, 21365200, 25744136, 30846024, 36763265, 43596840, 51456825, 60462921, 70744999, 82443660, 95710810, 110710250

Offset: 0

Wolfdieter Lang, Jul 28 2017

See A094816 and A290311.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A094816, A290311, A290312.

Vec((1 + 17*x - 2*x^2 - x^3) / (1 - x)^7 + O(x^50)) \\ Colin Barker, Jul 29 2017

O.g.f: (1 + 17*x - 2*x^2 - x^3)/(1 - x)^7.
E.g.f.: exp(x)*(1 + 23*x + 98*x^2/2! + 181*x^3/3! + 170*x^4/4! + 80*x^5/5! + 15*x^6/6!).
From Colin Barker, Jul 29 2017: (Start)
a(n) = (48 + 256*n + 422*n^2 + 303*n^3 + 105*n^4 + 17*n^5 + n^6) / 48.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 6.
(End)

A290314 Fifth diagonal sequence of the Sheffer triangle A094816 (special Charlier).

Original entry on oeis.org

1, 89, 814, 4179, 15659, 47775, 125853, 296703, 641058, 1290718, 2451449, 4432792, 7686042, 12851762, 20818302, 32792898, 50387031, 75717831, 111527416, 161322161, 229533997, 321705945, 444704195, 606959145, 818737920, 1092450996, 1442995659, 1888139134, 2448944324, 3150241204, 4021147020, 5095638548

Offset: 0

Wolfdieter Lang, Jul 28 2017

See A094816 and A290311.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A094816, A290311, A290312, A290313.

Vec((1 + 80*x + 49*x^2 - 27*x^3 + 2*x^4) / (1 - x)^9 + O(x^50)) \\ Colin Barker, Jul 29 2017

O.g.f.: (1  + 80*x + 49*x^2 - 27*x^3 + 2*x^4)/(1-x)^9.
E.g.f: exp(x)*(1 + 88*x + 637*x^2/2! + 2003*x^3/3! + 3472*x^4/4! + 3574*x^5/5! + 2185*x^6/6! + 735*x^7/7! + 105*x^8/8!).
From Colin Barker, Jul 29 2017: (Start)
a(n) = (5760 + 67248*n + 158180*n^2 + 161700*n^3 + 87695*n^4 + 26952*n^5 + 4670*n^6 + 420*n^7 + 15*n^8) / 5760.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8.
(End)

A290315 Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.

Original entry on oeis.org

1, 1, 2, 1, 16, 12, 1, 66, 284, 120, 1, 224, 2872, 5952, 1680, 1, 706, 21080, 116336, 146064, 30240, 1, 2160, 132228, 1531072, 4804656, 4130304, 665280, 1, 6530, 760500, 16271080, 101422640, 208791648, 132557760, 17297280, 1, 19648, 4155120, 151922560, 1661273440, 6556459008, 9657333504, 4766423040, 518918400, 1, 59010, 21993776, 1304454880, 23155279200, 155184721088, 427142449920, 477104352768, 189945688320, 17643225600

Offset: 0

Wolfdieter Lang, Jul 29 2017

The ordinary generating function (o.g.f.) of the (n+1)-th diagonal sequence of the Sheffer triangle A154537 = (e^x, e^(2*x) - 1), called S2[2,1], is GS2(2,1;n,x) = P(n, x)/(1 - 2*x)^(2*n+1), with the row polynomials P(n, x) = Sum_{k=0..n} T(n, k)*x^k, n >= 0.
In the general case of Sheffer S2[d,a] = (e^(a*x), e^(d*x) - 1) (with gcd(d,a) = 1, d >= 0, a >= 0, and for d = 1 one takes a = 0) the o.g.f. of the (n+1)-th diagonal sequence is G(d,a;n,x) = P(d,a;n,x)/(1 - d*x)^(2*n + 1) with the numerator polynomial P and coefficient table T(d,a;n,k).
For the computation of the exponential generating function (e.g.f.) of the o.g.f.s of the diagonal sequences of a Sheffer triangle (lower triangular matrix) via Lagrange's theorem see a comment in A290311.

			The triangle T(n, k) begins:
n\k  0    1      2        3         4         5         6        7 ...
0:   1
1:   1    2
2:   1   16     12
3:   1   66    284      120
4:   1  224   2872     5952      1680
5:   1  706  21080   116336    146064     30240
6:   1 2160 132228  1531072   4804656   4130304    665280
7:   1 6530 760500 16271080 101422640 208791648 132557760 17297280
...
n = 8: 1 19648 4155120 151922560 1661273440 6556459008 9657333504 4766423040 518918400,
n = 9: 1 59010 21993776 1304454880 23155279200 155184721088 427142449920 477104352768 189945688320 17643225600.
...
n=3: The o.g.f. of the 4th diagonal sequence of A154537, [1, 80, 1320, ...], is P(3, x) = (1 + 66*x + 284*x^2 + 120*x^3)/(1 - 2*x)^7.

Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], 2017.

Cf. A154537, A290311.

T(n, k) = [x^k] P(n, x) with the numerator polynomial in the o.g.f. of the (n+1)-th diagonal sequence of the triangle A154537. See a comment above.

A290316 Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.

Original entry on oeis.org

1, 1, 6, 1, 48, 90, 1, 234, 2214, 2160, 1, 996, 27432, 114588, 71280, 1, 4062, 260748, 2791800, 6770628, 2993760, 1, 16344, 2178630, 48256344, 280652364, 454137840, 152681760, 1, 65490, 16966530, 691711920, 7846782660, 29157089832, 34236464400, 9160905600, 1, 262092, 126820980, 8851303620, 174637926180, 1219804572672, 3187159638984, 2871984146400, 632102486400, 1, 1048518, 924701832, 105253405560, 3359003385600, 39425596747272, 188635513271256, 369150976563264, 265665182896800, 49303993939200

Offset: 0

Wolfdieter Lang, Aug 08 2017

The ordinary generating function (o.g.f.) of the (n+1)-th diagonal sequence of the Sheffer triangle A282629 = (e^x, e^(3*x) - 1), called S2[3,1], is  GS2(3,1;n,x) = P(n, x)/(1 - 3*x)^(2*n+1), with the row polynomials P(n, x) = Sum_{k=0..n} T(n, k)*x^k, n >= 0.
For the general case Sheffer S2[d,a] = (e^(a*x), e^(d*x) - 1) (with gcd(d,a) = 1, d >=0, a >= 0, and for d = 1 one takes a = 0) see a comment in A290315.
For the computation of the exponential generating function (e.g.f.) of the o.g.f.s of the diagonal sequences of a Sheffer triangle (lower triangular matrix) via Lagrange's theorem see a comment and link in A290311.

			The triangle T(n, k) begins:
n\k 0     1        2         3          4           5           6          7 ...
0:  1
1:  1     6
2:  1    48       90
3:  1   234     2214      2160
4:  1   996    27432    114588      71280
5:  1  4062   260748   2791800    6770628     2993760
6:  1 16344  2178630  48256344  280652364   454137840   152681760
7:  1 65490 16966530 691711920 7846782660 29157089832 34236464400 9160905600
...
n = 8: 1 262092 126820980 8851303620 174637926180 1219804572672 3187159638984 2871984146400 632102486400,
n = 9: 1 1048518 924701832 105253405560 3359003385600 39425596747272 188635513271256 369150976563264 265665182896800 49303993939200.
...
n = 3: The o.g.f. of the 4th diagonal sequence of A282629, [1, 255, 7380, ...], is P(3, x) = (1 + 234*x + 2214*x^2 + 2160*x^3)/(1 - 3*x)^7.

Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], 2017.

Cf. A282629, A290311, A290315.

T(n, k) = [x^k] P(n, x) with the numerator polynomials of the o.g.f. of the (n+1)-th diagonal sequence of the triangle A282629. See a comment above.

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A290312 Third diagonal sequence of the Sheffer triangle A094816 (special Charlier).

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A290313 Fourth diagonal sequence of the Sheffer triangle A094816 (special Charlier).

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A290314 Fifth diagonal sequence of the Sheffer triangle A094816 (special Charlier).

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A290315 Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.

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A290316 Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.

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