A154690 -id:A154690

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A174669

Sequence A154690 adjusted to leading one:t(n,m)=A154690(n,m)-A154690(n,0)+1

+20
0

1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 24, 32, 24, 1, 1, 58, 88, 88, 58, 1, 1, 140, 236, 256, 236, 140, 1, 1, 334, 628, 712, 712, 628, 334, 1, 1, 784, 1648, 1984, 1984, 1984, 1648, 784, 1, 1, 1810, 4240, 5536, 5536, 5536, 5536, 4240, 1810, 1, 1, 4116, 10676, 15296, 15776

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Row sums are:

1, 2, 6, 22, 82, 294, 1010, 3350, 10818, 34246, 106834,...

LINKS

Table of n, a(n) for n=0..59.

FORMULA

t(n,m)=A154690(n,m)-A154690(n,0)+1

EXAMPLE

{1},

{1, 1},

{1, 4, 1},

{1, 10, 10, 1},

{1, 24, 32, 24, 1},

{1, 58, 88, 88, 58, 1},

{1, 140, 236, 256, 236, 140, 1},

{1, 334, 628, 712, 712, 628, 334, 1},

{1, 784, 1648, 1984, 1984, 1984, 1648, 784, 1},

{1, 1810, 4240, 5536, 5536, 5536, 5536, 4240, 1810, 1},

{1, 4116, 10676, 15296, 15776, 15104, 15776, 15296, 10676, 4116, 1}

MATHEMATICA

a = 2; b = 1;

t[n_, m_] = (a^m*b^(n - m) + b^m*a^(n - m))*Binomial[n, m];

Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];

Flatten[%]

CROSSREFS

A154690(n, m)

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Mar 26 2010

STATUS

approved

A174673

Triangle read by rows: T(n,m)=A154694(n,m)-A154694(n,0)+1.

+10
1

1, 1, 1, 1, 36, 1, 1, 296, 296, 1, 1, 1932, 4656, 1932, 1, 1, 11696, 54086, 54086, 11696, 1, 1, 69048, 556596, 1042920, 556596, 69048, 1, 1, 405236, 5406866, 16866206, 16866206, 5406866, 405236, 1, 1, 2381700, 51004320, 247754256, 404837664

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Reduces the values in the triangle A154694 such that each row starts with 1.

Row sums are:

1, 2, 38, 594, 8522, 131566, 2294210, 45356618, 1007118218, 24839902470,

673894929842,...

LINKS

Table of n, a(n) for n=0..40.

FORMULA

t(n,m)=A154694(n,m)-A154694(n,0)+1

EXAMPLE

{1},

{1, 1},

{1, 36, 1},

{1, 296, 296, 1},

{1, 1932, 4656, 1932, 1},

{1, 11696, 54086, 54086, 11696, 1},

{1, 69048, 556596, 1042920, 556596, 69048, 1},

{1, 405236, 5406866, 16866206, 16866206, 5406866, 405236, 1},

{1, 2381700, 51004320, 247754256, 404837664, 247754256, 51004320, 2381700, 1},

{1, 14050376, 473595806, 3441231326, 8491073726, 8491073726, 3441231326, 473595806, 14050376, 1},

{1, 83216400, 4357421004, 46167420504, 164067684600, 244543444824, 164067684600, 46167420504, 4357421004, 83216400, 1}

MAPLE

A174673 := proc(n, m)

A154694(n, m)-A154694(n, 0)+1 ;

end proc:

seq(seq( A174673(n, m), m=0..n), n=0..10) ; # R. J. Mathar, Mar 11 2024

MATHEMATICA

Clear[t, p, q, n, m];

p = 2; q = 3;

t[n_, m_] = (p^(n - m)*q^m + p^m*q^( n - m))*Sum[(-1)^j*Binomial[n + 2, j]*(m - j + 1)^(n + 1), {j, 0, m + 1}];

Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];

Flatten[%]

CROSSREFS

A154690, A154692, A154693, A154964

KEYWORD

nonn,tabl,less

AUTHOR

Roger L. Bagula, Mar 26 2010

STATUS

approved

A174672

Sequence A154693 adjusted to leading one:t(n,m)=A154693(n,m)-A154693(n,0)+1

+10
0

1, 1, 1, 1, 12, 1, 1, 58, 58, 1, 1, 244, 512, 244, 1, 1, 994, 3592, 3592, 994, 1, 1, 4016, 23756, 38592, 23756, 4016, 1, 1, 16174, 154420, 374728, 374728, 154420, 16174, 1, 1, 65004, 993088, 3529104, 4997824, 3529104, 993088, 65004, 1, 1, 260842, 6314368

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Row sums are:

1, 2, 14, 118, 1002, 9174, 94138, 1090646, 14172218, 204490006, 3245253882,...

LINKS

Table of n, a(n) for n=0..47.

FORMULA

t(n,m)=A154693(n,m)-A154693(n,0)+1

EXAMPLE

{1},

{1, 1},

{1, 12, 1},

{1, 58, 58, 1},

{1, 244, 512, 244, 1},

{1, 994, 3592, 3592, 994, 1},

{1, 4016, 23756, 38592, 23756, 4016, 1},

{1, 16174, 154420, 374728, 374728, 154420, 16174, 1},

{1, 65004, 993088, 3529104, 4997824, 3529104, 993088, 65004, 1},

{1, 260842, 6314368, 32773312, 62896480, 62896480, 32773312, 6314368, 260842, 1},

{1, 1045480, 39684596, 299673344, 779048096, 1006350848, 779048096, 299673344, 39684596, 1045480, 1}

MATHEMATICA

Clear[t, p, q, n, m];

p = 2; q = 1;

t[n_, m_] = (p^(n - m)*q^m + p^m*q^( n - m))*Sum[(-1)^j*Binomial[n + 2, j]*(m - j + 1)^(n + 1), {j, 0, m + 1}];

Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];

Flatten[%]

CROSSREFS

A154690, A154692, A154693

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Mar 26 2010

STATUS

approved

A174674

Sequence A154695 adjusted to leading one:t(n,m)=A154695(n,m)-A154695(n,0)+1

+10
0

1, 1, 1, 1, 20, 1, 1, 130, 130, 1, 1, 744, 1824, 744, 1, 1, 4234, 20152, 20152, 4234, 1, 1, 24484, 210796, 376704, 210796, 24484, 1, 1, 143686, 2183524, 6233224, 6233224, 2183524, 143686, 1, 1, 851504, 22549360, 99411264, 149600192, 99411264

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Row sums are:

1, 2, 22, 262, 3314, 48774, 847266, 17120870, 395224450, 10263445126,

296140564130,...

LINKS

Table of n, a(n) for n=0..41.

FORMULA

t(n,m)=A154695(n,m)-A154695(n,0)+1

EXAMPLE

{1},

{1, 1},

{1, 20, 1},

{1, 130, 130, 1},

{1, 744, 1824, 744, 1},

{1, 4234, 20152, 20152, 4234, 1},

{1, 24484, 210796, 376704, 210796, 24484, 1},

{1, 143686, 2183524, 6233224, 6233224, 2183524, 143686, 1},

{1, 851504, 22549360, 99411264, 149600192, 99411264, 22549360, 851504, 1},

{1, 5075122, 231836368, 1562973472, 3331837600, 3331837600, 1562973472, 231836368, 5075122, 1},

{1, 30344508, 2370195636, 24248921920, 72553861536, 97733916928, 72553861536, 24248921920, 2370195636, 30344508, 1}

MATHEMATICA

Clear[t, p, q, n, m, a];

p[x_, n_] = 2^n*(1 - x)^(n + 1)*LerchPhi[x, -n, 1/2];

a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

p = 2; q = 1;

t[n_, m_] := (p^(n - m)*q^m + p^m*q^(n - m))*a[[n + 1]][[m + 1]];

Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];

Flatten[%]

CROSSREFS

A154690, A154692, A154693, A154694, A154695

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Mar 26 2010

STATUS

approved

A174675

Sequence A154696 adjusted to leading one:t(n,m)=A154696(n,m)-A154696(n,0)+1

+10
0

1, 1, 1, 1, 60, 1, 1, 656, 656, 1, 1, 5832, 16464, 5832, 1, 1, 49496, 302486, 302486, 49496, 1, 1, 419412, 4933332, 10171944, 4933332, 419412, 1, 1, 3593036, 76425506, 280498526, 280498526, 76425506, 3593036, 1, 1, 31167600, 1157982288

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Row sums are:

1, 2, 62, 1314, 28130, 703966, 20877434, 721034138, 28453293026,

1263142713270, 62305874244266,...

LINKS

Table of n, a(n) for n=0..38.

FORMULA

t(n,m)=A154696(n,m)-A154696(n,0)+1

EXAMPLE

{1},

{1, 1},

{1, 60, 1},

{1, 656, 656, 1},

{1, 5832, 16464, 5832, 1},

{1, 49496, 302486, 302486, 49496, 1},

{1, 419412, 4933332, 10171944, 4933332, 419412, 1},

{1, 3593036, 76425506, 280498526, 280498526, 76425506, 3593036, 1},

{1, 31167600, 1157982288, 6978681888, 12117629472, 6978681888, 1157982288, 31167600, 1},

{1, 273237776, 17387745806, 164112248126, 449798124926, 449798124926, 164112248126, 17387745806, 273237776, 1},

{1, 2414712204, 260247533196, 3735760480536, 15279843395064, 23749342002264, 15279843395064, 3735760480536, 260247533196, 2414712204, 1}

MATHEMATICA

Clear[t, p, q, n, m, a];

p[x_, n_] = 2^n*(1 - x)^(n + 1)*LerchPhi[x, -n, 1/2];

a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

p = 2; q = 3;

t[n_, m_] := (p^(n - m)*q^m + p^m*q^(n - m))*a[[n + 1]][[m + 1]];

Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];

Flatten[%]

CROSSREFS

A154690, A154692, A154693, A154694, A154695, A154696

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Mar 26 2010

STATUS

approved

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