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A284861
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Triangle read by rows: T(n, k) = S2[3,1](n, k)*k! with the Sheffer triangle S2[3,1] = (exp(x), exp(3*x) -1) given in A282629.
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7
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1, 1, 3, 1, 15, 18, 1, 63, 216, 162, 1, 255, 1890, 3564, 1944, 1, 1023, 14760, 52650, 68040, 29160, 1, 4095, 109458, 659340, 1516320, 1487160, 524880, 1, 16383, 790776, 7578522, 27624240, 46539360, 36741600, 11022480, 1, 65535, 5633730, 82902204, 450057384, 1158993360, 1535798880, 1014068160, 264539520
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OFFSET
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0,3
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COMMENTS
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This is a generalization of triangle A131689(n, k) = Stirling2(n, k)*k!, because S2[3,1] is a generalization of the Stirling2 triangle written as S2[1,0].
This triangle appears in the o.g.f. G(3,1;n,x) of the powers {(1+3*m)^n}_{m>=0} as G(3,1;n,x) = Sum_{k>=0..n} T(n, k)*x^k / (1-x)^k.
This triangle is also related to the generalized row reversed Euler triangle rEu[3,1] with row polynomial rEu(3,1;n,x) = Sum_{m=0..n} rEu(3,1;n,m)*x^m with rEu(3,1;n,m) = Sum_{j=0..m} (-1)^(m-j)*binomial(n-j, m-j)*T(n, m). This follows from the above given o.g.f. of powers G(3,1;n,x) = rEu(3,1;n,x)/(1-x)^(n+1). The Euler triangle E[3,1] (row reversed rEu[3,1] is given in A225117. See a formula below.
The e.g.f. of the row polynomials R(3,1;n,x) = Sum_{m=0..n} T(n, m)*x^m follows from the e.g.f. of the row polynomials of the Sheffer triangle A282629. See the formula section.
The diagonal sequence is A032031(k) = k!*3^k.
The row sums give unsigned A151919, and the alternating row sums give A122803.
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LINKS
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FORMULA
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E.g.f. of the row polynomials R(n, x) (see a comment above) is exp(z)/(1 - x*(exp(3*z) - 1)). This is the e.g.f. for the triangle.
T(n, k) = Sum_{m=0..k} binomial(k, m)*(-1)^(k-m)*(1+ 3*m)^n, 0 <= k <= n.
T(n, k) = Sum_{m=0..k} binomial(n-m, k-m)*A225117(n,n-m), 0 <= k <= n.
Three term recurrence: T(n, k) = 0 if n < k, T(n,-1) = 0, T(0, 0) = 1, T(n, k) = 3*k*T(n-1, k-1) + (1+3*k)*T(n-1, k) for n >= 1. See A282629.
The column k sequence has e.g.f. exp(x)*(exp(3*x) - 1)^k (from the Sheffer property of A282629).
The o.g.f. is A032031(k)*x^k/Product_{j=0..k} (1 - (1+3*j)*x).
n-th row polynomial R(n,x) = (1 + 3*x) o (1 + 3*x) o ... o (1 + 3*x) (n factors), where o denotes the black diamond multiplication operator of Dukes and White. See example E14 in the Bala link. Cf. A145901.
R(n,x) = Sum_{k = 0..n} binomial(n,k)*3^k*F(k,x) where F(k,x) is the Fubini polynomial of order k, the k-th row polynomial of A019538. (End)
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EXAMPLE
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The triangle T(n, k) begins
n\k 0 1 2 3 4 5 6 7 ...
0: 1
1: 1 3
2: 1 15 18
3: 1 63 216 162
4: 1 255 1890 3564 1944
5: 1 1023 14760 52650 68040 29160
6: 1 4095 109458 659340 1516320 1487160 524880
7: 1 16383 790776 7578522 27624240 46539360 36741600 11022480
...
row n=8: 1 65535 5633730 82902204 450057384 1158993360 1535798880 1014068160 264539520,
row n=9: 1 262143 39829320 879725610 6845572440 25294754520 50042059200 54561276000 30951123840 7142567040,
row n=10: 1 1048575 280378098 9155719980 99549149040 507399658920 1406104706160 2251231315200 2083248720000 1035672220800 214277011200.
------------------------------------------------------------------
T(2, 1) = -1 + 4^2 = 15 = 2*A225117(2,2) + 1*A225117(2,1) = 2*1 + 1*13.
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MATHEMATICA
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Table[Sum[Binomial[k, m] (-1)^(k - m) (1 + 3m)^n, {m, 0, k}], {n, 0, 10}, {k, 0, n}]// Flatten (* Indranil Ghosh, Apr 09 2017 *)
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PROG
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(PARI) for(n=0, 10, for(k=0, n, print1(sum(m=0, k, binomial(k, m) * (-1)^(k - m)*(1 + 3*m)^n), ", "); ); print(); ) \\ Indranil Ghosh, Apr 09 2017
from sympy import binomial
for n in range(11):
print([sum([binomial(k, m)*(-1)**(k - m)*(1 + 3*m)**n for m in range(k + 1)]) for k in range(n + 1)])
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CROSSREFS
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Cf. A000012, A002450, A016223, A021874, A032031, A111577, A122803, A131689, |A151919|, A225117, A282629, A019538, A145901.
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KEYWORD
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AUTHOR
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STATUS
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approved
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