{"id":"n-11124d93b13aca24","revision":0,"first_index":0,"fingerprint":"f6-cbe3741e35d7b661","name":"mobius_x_over_x_plus_one of Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","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"],["3","4"],["14","37","24"],["90","353","454","192"],["744","3846","7349","6166","1920"],["7560","48274","121771","151954","93936","23040"],["91440","692532","2161984","3566221","3281914","1599144","322560"],["1285200","11229756","41663532","85177609","103729396","75303388","30193584","5160960"],["20603520","203598000","873158012","2124586164","3210395649","3086846596","1845321372","627345072","92897280"],["371226240","4086822096","19857223188","55930492988","100701362091","120252606648","95283096868","48324697056","14239656000","1857945600"],["7428153600","90047769120","488235824856","1560078674628","3254995467338","4635520093821","4565029873788","3070646720428","1350531705456","350802570240","40874803200"],["163459296000","2161693196640","12924945773928","46141626786628","109325219683406","180573127745625","212225941128818","177530788046116","103611662439000","40188932779104","9325935509760","980995276800"],["3923502105600","56174272882560","366880183729344","1445899661280504","3831013498365052","7191390045503522","9809298307630821","9798704379104642","7115649280203892","3664089737073624","1270169115806304","266181669192960","25505877196800"],["102017281766400","1571289848588160","11122819144225920","47929492141402728","140311086149913108","294750811212731158","457216641931059011","530363994098908254","460137953038201868","294943240694622376","135785760385216800","42523448419245312","8120163046625280","714164561510400"],["2856571067750400","47077171894529280","358856455244356224","1677476207350311408","5373385526682100784","12479959495301828960","21676978392742856468","28610039816677074757","28837308063897518470","22092898355539497916","12665666061310558152","5269470621984599616","1504149177360205440","263703516402816000","21424936845312000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","75","1089","20025","446535","11715795","353743425","12084751665","460905128775","19414191656475","895154327661825","44844879530075625","2425592881954016775","140881250370301315875"],"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"],["3","4"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0"],["2","6","6","2"],["-2","-6","-6","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"],["3","6","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186292,"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-4e73ef2a3f6be861"]},{"id":186293,"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-4e73ef2a3f6be861"]},{"id":186294,"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-4e73ef2a3f6be861"]}],"region_exact":{},"incoming_edges":[{"id":"e-4e73ef2a3f6be861","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-11124d93b13aca24","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":"3fa01e182f1e0780e55c418e4cbe4fea600272479fe293f6ad42fd27d2b3aa08","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-6ace63f6e12c5046","role":"sequence","name":"Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","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-f80f9370395fda4c","kind":"fingerprint_related","node_a_id":"n-7a42f3124cafc007","node_b_id":"n-11124d93b13aca24","fingerprint":"f6-cbe3741e35d7b661","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cbe3741e35d7b661"},"other_id":"n-7a42f3124cafc007","other":{"id":"n-7a42f3124cafc007","name":"mobius_x_over_x_plus_one of Exponential Riordan array [1, log((1-x)/(1-2x))].","first_index":0,"fingerprint":"f6-cbe3741e35d7b661","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","3","4"]],"card_rows":[["1"],["0","1"],["0","3","4"],["0","14","37","24"]],"row_sums_preview":["1","1","7","75","1089","20025","446535","11715795"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e41c30a965e7cb9c","steps":["x \u21a6 x/(x+1)"],"edges":["e-b07c588e5f21b876"],"text":"A131222 \u2192 x \u21a6 x/(x+1)"},"headline":"A131222 \u2192 x \u21a6 x/(x+1)","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:         3,          4;\n  2:        14,         37,          24;\n  3:        90,        353,         454,         192;\n  4:       744,       3846,        7349,        6166,         1920;\n  5:      7560,      48274,      121771,      151954,        93936,        23040;\n  6:     91440,     692532,     2161984,     3566221,      3281914,      1599144,      322560;\n  7:   1285200,   11229756,    41663532,    85177609,    103729396,     75303388,    30193584,     5160960;\n  8:  20603520,  203598000,   873158012,  2124586164,   3210395649,   3086846596,  1845321372,   627345072,    92897280;\n  9: 371226240, 4086822096, 19857223188, 55930492988, 100701362091, 120252606648, 95283096868, 48324697056, 14239656000, 1857945600;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(2*n^2-2*n+(6*n^2-6*n)*x+(6*n^2-6*n)*x^2+(2*n^2-2*n)*x^3)*P(n-2,x) + (3*n+(6*n+1)*x+(3*n+1)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 2*n + (6*n^2 - 6*n)*x + (6*n^2 - 6*n)*x^2 + (2*n^2 - 2*n)*x^3)*P[n - 2, x] + (3*n + (6*n + 1)*x + (3*n + 1)*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; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 2*n + (6*n^2 - 6*n)*x + (6*n^2 - 6*n)*x^2 + (2*n^2 - 2*n)*x^3)*v[i - 2] + (3*n + (6*n + 1)*x + (3*n + 1)*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-6ace63f6e12c5046","steps":["x \u21a6 x/(x+1)"],"edges":["e-4e73ef2a3f6be861"],"text":"A079638 \u2192 x \u21a6 x/(x+1)"},"headline":"A079638 \u2192 x \u21a6 x/(x+1)","chain":null}
