{"id":"n-3e9ab13cc75c9520","revision":0,"first_index":0,"fingerprint":"f6-ce583c2d05188b3b","name":"mobius_x_over_x_plus_one of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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":true,"rows":[["1"],["0","1"],["0","1","2"],["0","4","13","10"],["0","36","157","220","100"],["0","576","3124","6189","5340","1700"],["0","14400","93076","235949","294414","181340","44200"],["0","518400","3883536","11937976","19329017","17421558","8300780","1635400"],["0","25401600","216213264","779137624","1544020633","1820107192","1277816120","495173600","81770000"],["0","1625702400","15488752896","63918670096","149461266072","216848201433","200087199160","114749158200","37419564000","5315050000"],["0","131681894400","1387896581376","6447490015248","17347693499704","29820528133977","33988615649466","25701832145320","12440415656400","3498923298000","435834100000"],["0","13168189440000","152089529472000","784926556243776","2385965841510448","4734169856867804","6410734906478277","6003033395128066","3839926612317320","1606374311096400","396974663098000","44019244100000"],["0","1593350922240000","20009352177792000","113531035901080896","384462906684504880","863922385345278940","1353267646221743605","1508476699400845780","1197001194296019772","662842338345377440","244011600188618800","53757237434056000","5370347780200000"],["0","229442532802560000","3112382597326848000","19249825235535489024","71824658768225432640","180151944958973374960","320139286930996525420","413444453415874615045","391097293391749485268","269014469894657218300","131249809487240836000","43122724217853790000","8568129508286920000","778700428129000000"],["0","38775788043632640000","564997889524672512000","3782325506351061805056","15410837621871131250240","42655870688664823917040","84729370134363886539180","124295791405552219264005","136380999663904347567942","111949985288794482388260","67913677685435428395000","29600208005648232630000","8778877003935633780000","1588182388762577400000","132379072781930000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","3","27","513","16929","863379","63026667","6239640033","804913564257","131200910973891","26371383105752091","6408246094697758113","1851983121367652094657","627822278143634060088723"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["1","3","2"],["-2","-4","-2"],["1","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185966,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-6c4346048c602f7b"]},{"id":185967,"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-6c4346048c602f7b"]},{"id":185968,"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-6c4346048c602f7b"]}],"region_exact":{},"incoming_edges":[{"id":"e-6c4346048c602f7b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-3e9ab13cc75c9520","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"5bd6a77f70e591c969318e44f88f57b16c68384c4e3ac212369849ae8c9ed0f0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-54284d24a094e2d3","role":"sequence","name":"Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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-65d5926c27201765","kind":"fingerprint_related","node_a_id":"n-3e9ab13cc75c9520","node_b_id":"n-6918d76cf43c5e24","fingerprint":"f6-ce583c2d05188b3b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ce583c2d05188b3b"},"other_id":"n-6918d76cf43c5e24","other":{"id":"n-6918d76cf43c5e24","name":"reversal of basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","first_index":0,"fingerprint":"f6-ce583c2d05188b3b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["4","13","10"]],"card_rows":[["1"],["1","2"],["4","13","10"],["36","157","220","100"]],"row_sums_preview":["1","3","27","513","16929","863379","63026667","6239640033"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e734a631f8279507","steps":["basis A007318","reverse"],"edges":["e-0b3719baa0dde23e","e-28343cb148a19d30"],"text":"A008955 \u2192 basis A007318 \u2192 reverse"},"headline":"A008955 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-875e80c6405a826b","kind":"fingerprint_related","node_a_id":"n-6a1d0871f44aed3c","node_b_id":"n-3e9ab13cc75c9520","fingerprint":"f6-ce583c2d05188b3b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ce583c2d05188b3b"},"other_id":"n-6a1d0871f44aed3c","other":{"id":"n-6a1d0871f44aed3c","name":"basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","first_index":0,"fingerprint":"f6-ce583c2d05188b3b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["10","13","4"]],"card_rows":[["1"],["2","1"],["10","13","4"],["100","220","157","36"]],"row_sums_preview":["1","3","27","513","16929","863379","63026667","6239640033"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e734a631f8279507","steps":["basis A007318"],"edges":["e-0b3719baa0dde23e"],"text":"A008955 \u2192 basis A007318"},"headline":"A008955 \u2192 basis A007318","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: 0,          1;\n  2: 0,          1,           2;\n  3: 0,          4,          13,          10;\n  4: 0,         36,         157,         220,          100;\n  5: 0,        576,        3124,        6189,         5340,         1700;\n  6: 0,      14400,       93076,      235949,       294414,       181340,        44200;\n  7: 0,     518400,     3883536,    11937976,     19329017,     17421558,      8300780,      1635400;\n  8: 0,   25401600,   216213264,   779137624,   1544020633,   1820107192,   1277816120,    495173600,    81770000;\n  9: 0, 1625702400, 15488752896, 63918670096, 149461266072, 216848201433, 200087199160, 114749158200, 37419564000, 5315050000;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n^2-2*n+1+(2*n^2-4*n+3)*x+(n^2-2*n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 - 2*n + 1 + (2*n^2 - 4*n + 3)*x + (n^2 - 2*n + 2)*x^2)*P[n - 1, x])/(1 + 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] = ((n^2 - 2*n + 1 + (2*n^2 - 4*n + 3)*x + (n^2 - 2*n + 2)*x^2)*v[i - 1])/(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-54284d24a094e2d3","steps":["x \u21a6 x/(x+1)"],"edges":["e-6c4346048c602f7b"],"text":"A269944 \u2192 x \u21a6 x/(x+1)"},"headline":"A269944 \u2192 x \u21a6 x/(x+1)","chain":null}
