{"id":"n-baf5264cc15878a7","revision":0,"first_index":0,"fingerprint":"f6-177c65847e3c5511","name":"basis_map_A013611 of derivative of Even-numbered rows of Pascal's triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[[],["4","8"],["32","192","384","256"],["192","1920","7680","15360","15360","6144"],["1024","14336","86016","286720","573440","688128","458752","131072"],["5120","92160","737280","3440640","10321920","20643840","27525120","23592960","11796480","2621440"],["24576","540672","5406720","32440320","129761280","363331584","726663168","1038090240","1038090240","692060160","276824064","50331648"],["114688","2981888","35782656","262406144","1312030720","4723310592","12595494912","25190989824","37786484736","41984983040","33587986432","18320719872","6106906624","939524096"],["524288","15728640","220200960","1908408320","11450449920","50381979648","167939932160","431845539840","863691079680","1343519457280","1612223348736","1465657589760","977105059840","450971566080","128849018880","17179869184"],["2359296","80216064","1283457024","12834570240","89841991680","467178356736","1868713426944","5873099341824","14682748354560","29365496709120","46984794734592","59798829662208","59798829662208","45999099740160","26285199851520","10514079940608","2628519985152","309237645312"],["10485760","398458880","7172259840","81285611520","650284892160","3901709352960","18207976980480","67629628784640","202888886353920","495950611087360","991901222174720","1623111090831360","2164148121108480","2330621053501440","1997675188715520","1331783459143680","665891729571840","235020610437120","52226802319360","5497558138880"],["46137344","1937768448","38755368960","490901340160","4418112061440","30043162017792","160230197428224","686700846120960","2403452961423360","6943308555223040","16663940532535296","33327881065070592","55546468441784320","76910494765547520","87897708303482880","82037861083250688","61528395812438016","36193174007316480","16085855114362880","5079743720325120","1015948744065024","96757023244288"],["201326592","9261023232","203742511104","2852395155456","28523951554560","216782031814656","1300692190887936","6317647784312832","25270591137251328","84235303790837760","235858850614345728","557484555997544448","1114969111995088896","1886870804914765824","2695529721306808320","3234635665568169984","3234635665568169984","2663817606938492928","1775878404625661952","934672844539822080","373869137815928832","106819753661693952","19421773393035264","1688849860263936"],["872415232","43620761600","1046898278400","16052440268800","176576842956800","1483245480837120","9888303205580800","53679360258867200","241557121164902400","912549124400742400","2920157198082375680","7964065085679206400","18582818533251481600","37165637066502963200","63712520685433651200","93445030338636021760","116806287923295027200","123677246036430028800","109935329810160025600","81004979860117913600","48602987916070748160","23144279960033689600","8416101803648614400","2195504818343116800","365917469723852800","29273397577908224"],["3758096384","202937204736","5276367323136","87939455385600","1055273464627200","9708515874570240","71195783080181760","427174698481090560","2135873492405452800","9018132523489689600","32465277084562882560","100347220079558000640","267592586878821335040","617521354335741542400","1235042708671483084800","2140740695030570680320","3211111042545856020480","4155555466824048967680","4617283852026721075200","4374268912446367334400","3499415129957093867520","2332943419971395911680","1272514592711670497280","553267214222465433600","184422404740821811200","44261377137797234688","6809442636584189952","504403158265495552"]],"degrees":[-1,1,3,5,7,9,11,13,15,17,19,21,23,25,27],"row_sums":["0","12","864","46656","2239488","100776960","4353564672","182849716224","7522959753216","304679870005248","12187194800209920","482612914088312832","18953525353286467584","739187488778172235776","28657730334169138987008"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["4","8"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["4","16","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":205215,"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-58f0eb5265d39366"]},{"id":205216,"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-58f0eb5265d39366"]},{"id":205217,"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-58f0eb5265d39366"]}],"region_exact":{},"incoming_edges":[{"id":"e-58f0eb5265d39366","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-baf5264cc15878a7","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"6c1fefef72daa67b60fa6bd87804d7979ebad660df19567c042cfecfa1d4449d","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-f1591709bf5cf41b","role":"sequence","name":"derivative of Even-numbered rows of Pascal's triangle.","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-36466cbac67b6a84","kind":"fingerprint_related","node_a_id":"n-91fbe90ea8777148","node_b_id":"n-baf5264cc15878a7","fingerprint":"f6-177c65847e3c5511","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-177c65847e3c5511"},"other_id":"n-91fbe90ea8777148","other":{"id":"n-91fbe90ea8777148","name":"shift_x_plus_one of derivative of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-177c65847e3c5511","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4","2"],["32","48","24","4"]],"card_rows":[["4","2"],["32","48","24","4"],["192","480","480","240","60","6"],["1024","3584","5376","4480","2240","672","112","8"]],"row_sums_preview":["0","6","108","1458","17496","196830","2125764","22320522"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["D","basis A007318"],"edges":["e-473cc94657c879cc","e-7f993f994f18d6ae"],"text":"A034870 \u2192 D \u2192 basis A007318"},"headline":"A034870 \u2192 D \u2192 basis A007318","chain":null}},{"id":"l-39d0c6a7a34e4c4f","kind":"fingerprint_related","node_a_id":"n-619cc8ea95273502","node_b_id":"n-baf5264cc15878a7","fingerprint":"f6-177c65847e3c5511","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-177c65847e3c5511"},"other_id":"n-619cc8ea95273502","other":{"id":"n-619cc8ea95273502","name":"derivative of basis_map_A013609 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-177c65847e3c5511","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4","8"],["8","48","96","64"]],"card_rows":[["4","8"],["8","48","96","64"],["12","120","480","960","960","384"],["16","224","1344","4480","8960","10752","7168","2048"]],"row_sums_preview":["0","12","216","2916","34992","393660","4251528","44641044"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["basis A013609","D"],"edges":["e-20c0b01c4fb345ed","e-c40d39309139e88c"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A013609 \u2192 D"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-5eb1eb9a621f0ea9","kind":"fingerprint_related","node_a_id":"n-c562f6d6978a6187","node_b_id":"n-baf5264cc15878a7","fingerprint":"f6-177c65847e3c5511","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-177c65847e3c5511"},"other_id":"n-c562f6d6978a6187","other":{"id":"n-c562f6d6978a6187","name":"derivative of basis_map_A007318 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-177c65847e3c5511","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["8","4"],["128","192","96","16"]],"card_rows":[["8","4"],["128","192","96","16"],["1536","3840","3840","1920","480","48"],["16384","57344","86016","71680","35840","10752","1792","128"]],"row_sums_preview":["0","12","432","11664","279936","6298560","136048896","2857026816"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["x \u21a6 x+1","D"],"edges":["e-9d7e9cd87d9b797c","e-22942c57873a064c"],"text":"A139548 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A139548 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-7052722f80f0f559","kind":"fingerprint_related","node_a_id":"n-0936e707d9d749c8","node_b_id":"n-baf5264cc15878a7","fingerprint":"f6-177c65847e3c5511","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-177c65847e3c5511"},"other_id":"n-0936e707d9d749c8","other":{"id":"n-0936e707d9d749c8","name":"basis_map_A013609 of derivative of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-177c65847e3c5511","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2","4"],["4","24","48","32"]],"card_rows":[["2","4"],["4","24","48","32"],["6","60","240","480","480","192"],["8","112","672","2240","4480","5376","3584","1024"]],"row_sums_preview":["0","6","108","1458","17496","196830","2125764","22320522"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["D","basis A013609"],"edges":["e-521a2a476289c028","e-ac0ca91da3703f90"],"text":"basis-map coefficients of A297191, shifted \u2192 D \u2192 basis A013609"},"headline":"basis-map coefficients of A297191, shifted \u2192 D \u2192 basis A013609","chain":null}},{"id":"l-b1692b2642569308","kind":"fingerprint_related","node_a_id":"n-cb21c99be919a143","node_b_id":"n-baf5264cc15878a7","fingerprint":"f6-177c65847e3c5511","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-177c65847e3c5511"},"other_id":"n-cb21c99be919a143","other":{"id":"n-cb21c99be919a143","name":"derivative of basis_map_A013611 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-177c65847e3c5511","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["16","32"],["128","768","1536","1024"]],"card_rows":[["16","32"],["128","768","1536","1024"],["768","7680","30720","61440","61440","24576"],["4096","57344","344064","1146880","2293760","2752512","1835008","524288"]],"row_sums_preview":["0","48","3456","186624","8957952","403107840","17414258688","731398864896"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013611","D"],"edges":["e-a6b5412d61e1c4e3","e-ede1d0c7432adab9"],"text":"A034870 \u2192 basis A013611 \u2192 D"},"headline":"A034870 \u2192 basis A013611 \u2192 D","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:       0;\n  1:       4,        8;\n  2:      32,      192,        384,         256;\n  3:     192,     1920,       7680,       15360,       15360,         6144;\n  4:    1024,    14336,      86016,      286720,      573440,       688128,        458752,        131072;\n  5:    5120,    92160,     737280,     3440640,    10321920,     20643840,      27525120,      23592960,       11796480,        2621440;\n  6:   24576,   540672,    5406720,    32440320,   129761280,    363331584,     726663168,    1038090240,     1038090240,      692060160,      276824064,       50331648;\n  7:  114688,  2981888,   35782656,   262406144,  1312030720,   4723310592,   12595494912,   25190989824,    37786484736,    41984983040,    33587986432,    18320719872,     6106906624,      939524096;\n  8:  524288, 15728640,  220200960,  1908408320, 11450449920,  50381979648,  167939932160,  431845539840,   863691079680,  1343519457280,  1612223348736,  1465657589760,   977105059840,   450971566080,   128849018880,    17179869184;\n  9: 2359296, 80216064, 1283457024, 12834570240, 89841991680, 467178356736, 1868713426944, 5873099341824, 14682748354560, 29365496709120, 46984794734592, 59798829662208, 59798829662208, 45999099740160, 26285199851520, 10514079940608, 2628519985152, 309237645312;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (4*n+16*n*x+16*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4 + 8*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n + 16*n*x + 16*n*x^2)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 16*n*x + 16*n*x^2)*v[i - 1])/(n - 1)); 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-08f49539099949d5","steps":["D","basis A013611"],"edges":["e-473cc94657c879cc","e-58f0eb5265d39366"],"text":"A034870 \u2192 D \u2192 basis A013611"},"headline":"A034870 \u2192 D \u2192 basis A013611","chain":null}
