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A171738
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Number of n-digit terms in A115853.
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1
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0, 9, 9, 252, 819, 11754, 72585, 803448, 6978159, 73047510, 744922341, 8023947732, 88219609227, 993117723282, 11397388906305, 132852212160624, 1568346473860839, 18699577205645646, 224600363892164061, 2711096523623447820, 32815659723020049411, 397495008150096639114
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history;
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OFFSET
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1,2
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (220, -23595, 1644214, -83716633, 3320655624, -106839372783, 2866614688938, -65446367283297, 1290934186719996, -22263630922575471, 338916526924263606, -4589394288200909781, 55636105851803049936, -607029027026607279171, 5987676299284026149946, -53597962913874542531511, 436793267723498418791916, -3249590798297591166114241, 22121531283842396531966386, -138066798468560248794910299, 791349936276906779789655496, -4171090447580882317650845341, 20240413531917813060282288558, -90503364704585154511917229755, 373146584463992545811390561028, -1419285864685331116742908160565, 4981481723295791035938228563970, -16135667263673226340044156972087, 48229932136237067328168799757664, -132993220541110610037595712319729, 338163168478885883461275641450094, -792363490141660083452341647812844, 1709441117632380221064502861863864, -3391977966130763237758648740886704, 6182428869770409717691044568257504, -10334792538707263177934951305545664, 15815751917789533624508526223437184, -22110429327513201841008428041853184, 28167208461596745245212310681342464, -32604111806052111009853487965004800, 34175732364859420044995467888502784, -32312338074096507084007090667556864, 27429278013461965544387449830678528, -20791237185298123019559840061734912, 13980827805208637058110391660085248, -8274715534505360843464786947735552, 4269275618057660740303251391905792, -1897199932751339113592691061948416, 715083586820263550683946460119040, -224022790650071250708428803276800, 56733883860143782469881036800000, -11154067599888012976481894400000, 1596567281510638039125196800000, -147945254464527104212992000000, 6658606584104736522240000000).
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FORMULA
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a(n) = Sum_{k=0..9} k!*C(9,k)*(S_2(n,k) + k*S_2(n,k+1)) = 9*Sum_{k=0..9} k!*C(9,k)*S_2(n,k+1), where S_2(n,k) = A008299(n,k).
a(n) = 9*Sum_{k=0..10} (-1)^k*9!/(10-k)!*C(n,k)*(10-k)^(n-k) for n >= 10, where 0^0 = 1.
All terms from a(11) onward satisfy a linear recurrence with characteristic polynomial (1-x)^10*(2-x)^9*(3-x)^8*(4-x)^7*(5-x)^6*(6-x)^5*(7-x)^4*(8-x)^3*(9-x)^2*(10-x). (End)
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MATHEMATICA
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Table[9 Sum[k! Binomial[9, k] (-1)^i Binomial[n, i] StirlingS2[n - i, k - i + 1], {k, 0, 9}, {i, 0, Min[n, k + 1]}], {n, 21}] (* Matthew House, Sep 06 2020 *)
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CROSSREFS
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Cf. A115853 (numbers where every present decimal digit occurs more than once).
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KEYWORD
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base,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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