{"id":"n-4b24cbbd5a9a86e3","revision":0,"first_index":0,"fingerprint":"f6-8fea465cea2922a0","name":"basis_map_A008277 of basis_map_A013611 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["8","4"],["96","96","16"],["1536","2304","768","64"],["30720","61440","30720","5120","256"],["737280","1843200","1228800","307200","30720","1024"],["20643840","61931520","51609600","17203200","2580480","172032","4096"],["660602880","2312110080","2312110080","963379200","192675840","19267584","917504","16384"],["23781703680","95126814720","110981283840","55490641920","13872660480","1849688064","132120576","4718592","65536"],["951268147200","4280706662400","5707608883200","3329438515200","998831554560","166471925760","15854469120","849346560","23592960","262144"],["41855798476800","209278992384000","313918488576000","209278992384000","73247647334400","14649529466880","1743991603200","124570828800","5190451200","115343360","1048576"],["2009078326886400","11049930797875200","18416551329792000","13812413497344000","5524965398937600","1289158593085440","184165513297920","16443349401600","913519411200","30450647040","553648128","4194304"],["104472072998092800","626832437988556800","1149192802979020800","957660669149184000","430947301117132800","114919280297902080","19153213382983680","2052130005319680","142509028147200","6333734584320","172738215936","2617245696","16777216"],["5850436087893196800","38027834571305779200","76055669142611558400","69717696714060595200","34858848357030297600","10457654507109089280","1991934191830302720","248991773978787840","20749314498232320","1152739694346240","41917807067136","952677433344","12213813248","67108864"],["351026165273591808000","2457183156915142656000","5323896839982809088000","5323896839982809088000","2928143261990544998400","976047753996848332800","209153090142181785600","29879012877454540800","2904904029752524800","193660268650168320","8802739484098560","266749681336320","5129801564160","56371445760","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","208","4672","128256","4148224","154144768","6461079552","301239697408","15451053359104","864103317897216","52303651334520832","3405378926053163008","237231010282777608192","17602333760135391870976"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","4"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197857,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-63582ef6c863bc64"]},{"id":197858,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-63582ef6c863bc64"]},{"id":197859,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-63582ef6c863bc64"]}],"region_exact":{},"incoming_edges":[{"id":"e-63582ef6c863bc64","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-4b24cbbd5a9a86e3","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"b4be6557f8189422a6c3801d5b12a7e7f0cc8a48e53f8e9476a32219ac738da2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f4e656360cd62235","role":"sequence","name":"basis_map_A013611 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-1b86caf62af85e74","kind":"fingerprint_related","node_a_id":"n-d55979c30b97975a","node_b_id":"n-4b24cbbd5a9a86e3","fingerprint":"f6-8fea465cea2922a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8fea465cea2922a0"},"other_id":"n-d55979c30b97975a","other":{"id":"n-d55979c30b97975a","name":"Triangular array A066667 or A008297 unsigned and transposed.","first_index":0,"fingerprint":"f6-8fea465cea2922a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","12","36","24"]],"row_sums_preview":["1","3","13","73","501","4051","37633","394353"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A089231"],"derivation":{"root":"n-d55979c30b97975a","steps":[],"edges":[],"text":"A089231"},"headline":"A089231: Triangular array A066667 or A008297 unsigned and transposed.","chain":null}},{"id":"l-50cc2eeae6183a59","kind":"fingerprint_related","node_a_id":"n-44f00d1b1bfc267d","node_b_id":"n-4b24cbbd5a9a86e3","fingerprint":"f6-8fea465cea2922a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8fea465cea2922a0"},"other_id":"n-44f00d1b1bfc267d","other":{"id":"n-44f00d1b1bfc267d","name":"reversal of Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.","first_index":1,"fingerprint":"f6-8fea465cea2922a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","12","36","24"]],"row_sums_preview":["1","3","13","73","501","4051","37633","394353"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4a0fd05ab0621175","steps":["reverse"],"edges":["e-9d67892de8bd5f06"],"text":"A105278 \u2192 reverse"},"headline":"A105278 \u2192 reverse","chain":null}},{"id":"l-5678779e577e7ba7","kind":"fingerprint_related","node_a_id":"n-74cc871ad78dd737","node_b_id":"n-4b24cbbd5a9a86e3","fingerprint":"f6-8fea465cea2922a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8fea465cea2922a0"},"other_id":"n-74cc871ad78dd737","other":{"id":"n-74cc871ad78dd737","name":"basis_map_A008277 of basis_map_A013610 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","first_index":0,"fingerprint":"f6-8fea465cea2922a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","3"],["54","54","9"]],"card_rows":[["1"],["6","3"],["54","54","9"],["648","972","324","27"]],"row_sums_preview":["1","9","117","1971","40581","984393","27434457","862450011"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8229dbdd6a4b5cd9","steps":["basis A013610","basis A008277"],"edges":["e-86f8b58a2aa07621","e-e8d378d7ecb33a08"],"text":"A225470 \u2192 basis A013610 \u2192 basis A008277"},"headline":"A225470 \u2192 basis A013610 \u2192 basis A008277","chain":null}},{"id":"l-5c3cd50b58d52209","kind":"fingerprint_related","node_a_id":"n-011d67b73ad91983","node_b_id":"n-4b24cbbd5a9a86e3","fingerprint":"f6-8fea465cea2922a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8fea465cea2922a0"},"other_id":"n-011d67b73ad91983","other":{"id":"n-011d67b73ad91983","name":"basis_map_A008277 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","first_index":0,"fingerprint":"f6-8fea465cea2922a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["6","6","1"]],"card_rows":[["1"],["2","1"],["6","6","1"],["24","36","12","1"]],"row_sums_preview":["1","3","13","73","501","4051","37633","394353"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["basis A008277"],"edges":["e-31e8b29857f291e5"],"text":"A130534 \u2192 basis A008277"},"headline":"A130534 \u2192 basis A008277","chain":null}},{"id":"l-5c41661917eb7b10","kind":"fingerprint_related","node_a_id":"n-9e4edf560ed848c1","node_b_id":"n-4b24cbbd5a9a86e3","fingerprint":"f6-8fea465cea2922a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8fea465cea2922a0"},"other_id":"n-9e4edf560ed848c1","other":{"id":"n-9e4edf560ed848c1","name":"Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.","first_index":1,"fingerprint":"f6-8fea465cea2922a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","2","1"]],"card_rows":[["1"],["0","1"],["0","2","1"],["0","6","6","1"]],"row_sums_preview":["1","1","3","13","73","501","4051","37633"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A271703"],"derivation":{"root":"n-9e4edf560ed848c1","steps":[],"edges":[],"text":"A271703"},"headline":"A271703: Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.","chain":null}},{"id":"l-614270dec7307ec0","kind":"fingerprint_related","node_a_id":"n-4a0fd05ab0621175","node_b_id":"n-4b24cbbd5a9a86e3","fingerprint":"f6-8fea465cea2922a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8fea465cea2922a0"},"other_id":"n-4a0fd05ab0621175","other":{"id":"n-4a0fd05ab0621175","name":"Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.","first_index":1,"fingerprint":"f6-8fea465cea2922a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["6","6","1"]],"card_rows":[["1"],["2","1"],["6","6","1"],["24","36","12","1"]],"row_sums_preview":["1","3","13","73","501","4051","37633","394353"],"regions":["(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":["A105278"],"derivation":{"root":"n-4a0fd05ab0621175","steps":[],"edges":[],"text":"A105278"},"headline":"A105278: Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.","chain":null}},{"id":"l-783c30bc0ceebd7e","kind":"fingerprint_related","node_a_id":"n-742f536f1f5b6b99","node_b_id":"n-4b24cbbd5a9a86e3","fingerprint":"f6-8fea465cea2922a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8fea465cea2922a0"},"other_id":"n-742f536f1f5b6b99","other":{"id":"n-742f536f1f5b6b99","name":"derivative of basis_map_A008277 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-8fea465cea2922a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["4","2"]],"card_rows":[["1"],["4","2"],["18","18","3"],["96","144","48","4"]],"row_sums_preview":["0","1","6","39","292","2505","24306","263431"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["basis A008277","D"],"edges":["e-07cfebc08bea7e3c","e-51f0a866245c3734"],"text":"A132393 \u2192 basis A008277 \u2192 D"},"headline":"A132393 \u2192 basis A008277 \u2192 D","chain":null}},{"id":"l-9666226573698a69","kind":"fingerprint_related","node_a_id":"n-01c7db485d232340","node_b_id":"n-4b24cbbd5a9a86e3","fingerprint":"f6-8fea465cea2922a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8fea465cea2922a0"},"other_id":"n-01c7db485d232340","other":{"id":"n-01c7db485d232340","name":"derivative of Triangular array A066667 or A008297 unsigned and transposed.","first_index":0,"fingerprint":"f6-8fea465cea2922a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["6","12"]],"card_rows":[["2"],["6","12"],["12","72","72"],["20","240","720","480"]],"row_sums_preview":["0","2","18","156","1460","15030","170142","2107448"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d55979c30b97975a","steps":["D"],"edges":["e-f4173f0617a4e16b"],"text":"A089231 \u2192 D"},"headline":"A089231 \u2192 D","chain":null}},{"id":"l-c224b27416263d53","kind":"fingerprint_related","node_a_id":"n-d33444174e99cf8f","node_b_id":"n-4b24cbbd5a9a86e3","fingerprint":"f6-8fea465cea2922a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8fea465cea2922a0"},"other_id":"n-d33444174e99cf8f","other":{"id":"n-d33444174e99cf8f","name":"reversal of Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.","first_index":1,"fingerprint":"f6-8fea465cea2922a0","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","2"]],"card_rows":[["1"],["1"],["1","2"],["1","6","6"]],"row_sums_preview":["1","1","3","13","73","501","4051","37633"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9e4edf560ed848c1","steps":["reverse"],"edges":["e-38579b481f3726b6"],"text":"A271703 \u2192 reverse"},"headline":"A271703 \u2192 reverse","chain":null}},{"id":"l-f95d6848a2292654","kind":"fingerprint_related","node_a_id":"n-63664dd43379985f","node_b_id":"n-4b24cbbd5a9a86e3","fingerprint":"f6-8fea465cea2922a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8fea465cea2922a0"},"other_id":"n-63664dd43379985f","other":{"id":"n-63664dd43379985f","name":"basis_map_A008277 of basis_map_A013609 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-8fea465cea2922a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["24","24","4"]],"card_rows":[["1"],["4","2"],["24","24","4"],["192","288","96","8"]],"row_sums_preview":["1","6","52","584","8016","129632","2408512","50477184"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A013609","basis A008277"],"edges":["e-03658933e548767a","e-fe25af3288e5f765"],"text":"A028338 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A028338 \u2192 basis A013609 \u2192 basis A008277","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:            1;\n  1:            8,             4;\n  2:           96,            96,            16;\n  3:         1536,          2304,           768,            64;\n  4:        30720,         61440,         30720,          5120,          256;\n  5:       737280,       1843200,       1228800,        307200,        30720,         1024;\n  6:     20643840,      61931520,      51609600,      17203200,      2580480,       172032,        4096;\n  7:    660602880,    2312110080,    2312110080,     963379200,    192675840,     19267584,      917504,     16384;\n  8:  23781703680,   95126814720,  110981283840,   55490641920,  13872660480,   1849688064,   132120576,   4718592,    65536;\n  9: 951268147200, 4280706662400, 5707608883200, 3329438515200, 998831554560, 166471925760, 15854469120, 849346560, 23592960, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+4+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4*n + 4 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4*n + 4 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-9c5c208df8605708","steps":["basis A013611","basis A008277"],"edges":["e-8a4919ac2dd1c88f","e-63582ef6c863bc64"],"text":"A225471 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A225471 \u2192 basis A013611 \u2192 basis A008277","chain":null}
