{"id":"n-48b0227ab432072e","revision":0,"first_index":0,"fingerprint":"f6-b2b1f354adf1851f","name":"mobius_x_over_x_plus_one of basis_map_A013610 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.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["8","40","50"],["64","640","2400","4000","2500"],["512","7680","48000","160000","300000","300000","125000"],["4096","81920","716800","3584000","11200000","22400000","28000000","20000000","6250000"],["32768","819200","9216000","61440000","268800000","806400000","1680000000","2400000000","2250000000","1250000000","312500000"],["262144","7864320","108134400","901120000","5068800000","20275200000","59136000000","126720000000","198000000000","220000000000","165000000000","75000000000","15625000000"],["2097152","73400320","1192755200","11927552000","82001920000","410009600000","1537536000000","4392960000000","9609600000000","16016000000000","20020000000000","18200000000000","11375000000000","4375000000000","781250000000"],["16777216","671088640","12582912000","146800640000","1192755200000","7156531200000","32800768000000","117145600000000","329472000000000","732160000000000","1281280000000000","1747200000000000","1820000000000000","1400000000000000","750000000000000","250000000000000","39062500000000"],["134217728","6039797760","128345702400","1711276032000","16043212800000","112302489600000","608305152000000","2607022080000000","8961638400000000","24893440000000000","56010240000000000","101836800000000000","148512000000000000","171360000000000000","153000000000000000","102000000000000000","47812500000000000","14062500000000000","1953125000000000"],["1073741824","53687091200","1275068416000","19126026240000","203214028800000","1625712230400000","10160701440000000","50803507200000000","206389248000000000","687964160000000000","1891901440000000000","4299776000000000000","8062080000000000000","12403200000000000000","15504000000000000000","15504000000000000000","12112500000000000000","7125000000000000000","2968750000000000000","781250000000000000","97656250000000000"],["8589934592","472446402560","12401718067200","206695301120000","2454506700800000","22090560307200000","156474802176000000","894141726720000000","4191289344000000000","16299458560000000000","52973240320000000000","144472473600000000000","331082752000000000000","636697600000000000000","1023264000000000000000","1364352000000000000000","1492260000000000000000","1316700000000000000000","914375000000000000000","481250000000000000000","180468750000000000000","42968750000000000000","4882812500000000000"],["68719476736","4123168604160","118541097369600","2173253451776000","28523951554560000","285239515545600000","2258146164736000000","14516653916160000000","77119723929600000000","342754328576000000000","1285328732160000000000","4089682329600000000000","11076222976000000000000","25560514560000000000000","50208153600000000000000","83680256000000000000000","117675360000000000000000","138441600000000000000000","134596000000000000000000","106260000000000000000000","66412500000000000000000","31625000000000000000000","10781250000000000000000","2343750000000000000000","244140625000000000000"],["549755813888","35734127902720","1116691496960000","22333829939200000","321048805376000000","3531536859136000000","30900947517440000000","220721053696000000000","1310531256320000000000","6552656281600000000000","27848789196800000000000","101268324352000000000000","316463513600000000000000","852017152000000000000000","1977896960000000000000000","3955793920000000000000000","6799020800000000000000000","9998560000000000000000000","12498200000000000000000000","13156000000000000000000000","11511500000000000000000000","8222500000000000000000000","4671875000000000000000000","2031250000000000000000000","634765625000000000000000","126953125000000000000000","12207031250000000000000"],["4398046511104","307863255777280","10390384882483200","225125005787136000","3517578215424000000","42210938585088000000","404521494773760000000","3178383173222400000000","20858139574272000000000","115878553190400000000000","550423127654400000000000","2251730976768000000000000","7974880542720000000000000","24538093977600000000000000","65727037440000000000000000","153363087360000000000000000","311518771200000000000000000","549739008000000000000000000","839879040000000000000000000","1105104000000000000000000000","1243242000000000000000000000","1184040000000000000000000000","941850000000000000000000000","614250000000000000000000000","319921875000000000000000000","127968750000000000000000000","36914062500000000000000000","6835937500000000000000000","610351562500000000000000"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","98","9604","941192","92236816","9039207968","885842380864","86812553324672","8507630225817856","833747762130149888","81707280688754689024","8007313507497959524352","784716723734800033386496","76902238926010403271876608","7536419414749019520643907584"],"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","2","1"]]},"terms":[{"coeff":{"coeffs":[["8","56","138","140","50"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193637,"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-e668926f5ed327ee"]},{"id":193638,"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-e668926f5ed327ee"]},{"id":193639,"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-e668926f5ed327ee"]}],"region_exact":{},"incoming_edges":[{"id":"e-e668926f5ed327ee","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\":2}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-48b0227ab432072e","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":"8131c5ca87a16eb2a036410be5b7241c68d2fba54ac95d9827b2816d2ca70991","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-87b7b64af43d2638","role":"sequence","name":"basis_map_A013610 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.","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-123751ae14eecd28","kind":"fingerprint_related","node_a_id":"n-2d2986fb6c657dd8","node_b_id":"n-48b0227ab432072e","fingerprint":"f6-b2b1f354adf1851f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b2b1f354adf1851f"},"other_id":"n-2d2986fb6c657dd8","other":{"id":"n-2d2986fb6c657dd8","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-b2b1f354adf1851f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","20","25"],["16","160","600","1000","625"]],"card_rows":[["1"],["4","20","25"],["16","160","600","1000","625"],["64","960","6000","20000","37500","37500","15625"]],"row_sums_preview":["1","49","2401","117649","5764801","282475249","13841287201","678223072849"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-cb7f9b19b94bd36e","e-84daae9eebeef2d2"],"text":"A034870 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A034870 \u2192 basis A013610 \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:         8,         40,           50;\n  2:        64,        640,         2400,          4000,           2500;\n  3:       512,       7680,        48000,        160000,         300000,          300000,          125000;\n  4:      4096,      81920,       716800,       3584000,       11200000,        22400000,        28000000,         20000000,          6250000;\n  5:     32768,     819200,      9216000,      61440000,      268800000,       806400000,      1680000000,       2400000000,       2250000000,        1250000000,         312500000;\n  6:    262144,    7864320,    108134400,     901120000,     5068800000,     20275200000,     59136000000,     126720000000,     198000000000,      220000000000,      165000000000,        75000000000,        15625000000;\n  7:   2097152,   73400320,   1192755200,   11927552000,    82001920000,    410009600000,   1537536000000,    4392960000000,    9609600000000,    16016000000000,    20020000000000,     18200000000000,     11375000000000,      4375000000000,       781250000000;\n  8:  16777216,  671088640,  12582912000,  146800640000,  1192755200000,   7156531200000,  32800768000000,  117145600000000,  329472000000000,   732160000000000,  1281280000000000,   1747200000000000,   1820000000000000,   1400000000000000,    750000000000000,    250000000000000,    39062500000000;\n  9: 134217728, 6039797760, 128345702400, 1711276032000, 16043212800000, 112302489600000, 608305152000000, 2607022080000000, 8961638400000000, 24893440000000000, 56010240000000000, 101836800000000000, 148512000000000000, 171360000000000000, 153000000000000000, 102000000000000000, 47812500000000000, 14062500000000000, 1953125000000000;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (8+56*x+138*x^2+140*x^3+50*x^4)*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[((8 + 56*x + 138*x^2 + 140*x^3 + 50*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];\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] = ((8 + 56*x + 138*x^2 + 140*x^3 + 50*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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-336459bdce627a87","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-a142fb8711dcbce4","e-e668926f5ed327ee"],"text":"A139548 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A139548 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}
