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FORMULA
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It is also possible to calculate these coefficients using this formula: a(k,3,p) = Sum_{i=0..k-3}(-1)^i*binomial(3*p+1,i)*binomial(k-i,3)^p ; (k=3+i).
a(3,3,p) = 1; a(4,3,p) = 4^p - (3*p+1); a(5,3,p) = 10^p - (3*p+1)*4^p + binomial(3*p+1,2); a(6,3,p) = 20^p - (3*p+1)*10^p + binomial(3*p+1,2)*4^p - binomial(3*p+1,3);
With these coefficients we can calculate: Sum_{i=1..n}binomial(2+i,3)^p = Sum_{i=0..3*p-3} a(3+i,3,p)*binomial(n+3+i,3*p+1).
Example: Sum_{i=0..10}binomial(2+i,3)^4 = binomial(13,13) + 243*binomial(14,13) + 6750*binomial(15,13) + 49682*binomial(16,13) + 128124*binomial(17,13) + 128124*binomial(18,13) + 49682*binomial(19,13) + 6750*binomial(20,13) + 243*binomial(21,13) + binomial(22,13) = 3352413139. [Yahia Kahloune, Jan 30 2014].
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EXAMPLE
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{1},
{1, 9, 9, 1},
{1, 54, 405, 760, 405, 54, 1},
{1, 243, 6750, 49682, 128124, 128124, 49682, 6750, 243, 1},
{1, 1008, 83736, 1722320, 12750255, 40241088, 58571184, 40241088, 12750255, 1722320, 83736, 1008, 1},
{1, 4077, 922347, 45699447, 789300477, 5904797049, 21475242671, 40396577931, 40396577931, 21475242671, 5904797049, 789300477, 45699447, 922347, 4077, 1},
{1, 16362, 9639783, 1063783164, 38464072830, 592030140912, 4476844162434, 18096792917796, 41106807537048, 53885342499340, 41106807537048, 18096792917796, 4476844162434, 592030140912, 38464072830, 1063783164, 9639783, 16362, 1},
{1, 65511, 98361900, 23119658500, 1641724670475, 47871255785661, 678770257169016, 5183615502649800, 22745757394235250, 59751188387945950, 96290611703937936, 96290611703937936, 59751188387945950, 22745757394235250, 5183615502649800, 678770257169016, 47871255785661, 1641724670475, 23119658500, 98361900, 65511, 1},
{1, 262116, 992660346, 484099087156, 64856779908606, 3399596932632516, 84698452637705746, 1129236431002624116, 8699569720553953791, 40765121565728774056, 120242049753585134196, 228223252150517271816, 282197168724547971076, 228223252150517271816, 120242049753585134196, 40765121565728774056, 8699569720553953791, 1129236431002624116, 84698452637705746, 3399596932632516, 64856779908606, 484099087156, 992660346, 262116, 1},
{1, 1048545, 9967494609, 9930487583345, 2445752640197970, 222507204130403730, 9324662905839457490, 205937718403143468690, 2617057246555282014495, 20317575263352346466495, 100469410825316110531695, 325788868009936985275215, 706545050062795671491900, 1037445219390759634565820, 1037445219390759634565820, 706545050062795671491900, 325788868009936985275215, 100469410825316110531695, 20317575263352346466495, 2617057246555282014495, 205937718403143468690, 9324662905839457490, 222507204130403730, 2445752640197970, 9930487583345, 9967494609, 1048545, 1}
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