{"id":"n-af48e57b5ab0efc6","revision":0,"first_index":0,"fingerprint":"f6-6b852c5cd7d4c499","name":"reversal 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"],["18","24","8"],["324","864","864","384","64"],["5832","23328","38880","34560","17280","4608","512"],["104976","559872","1306368","1741824","1451520","774144","258048","49152","4096"],["1889568","12597120","37791360","67184640","78382080","62705664","34836480","13271040","3317760","491520","32768"],["34012224","272097792","997691904","2217093120","3325639680","3547348992","2759049216","1576599552","656916480","194641920","38928384","4718592","262144"],["612220032","5714053632","24760899072","66029064192","121053284352","161404379136","161404379136","122974765056","71735279616","31882346496","10627448832","2576351232","429391872","44040192","2097152"],["11019960576","117546246144","587731230720","1828497162240","3961743851520","6338790162432","7747410198528","7378485903360","5533864427520","3279327068160","1530352631808","556491866112","154581073920","31708938240","4529848320","402653184","16777216"],["198359290368","2380311484416","13488431745024","47958868426752","119897171066880","223808052658176","323278298284032","369460912324608","338672502964224","250868520714240","150521112428544","72979933298688","28381085171712","8732641591296","2079200378880","369635622912","46204452864","3623878656","134217728"],["3570467226624","47606229688320","301506121359360","1206024485437440","3417069375406080","7289748000866304","12149580001443840","16199440001925120","17549393335418880","15599460742594560","11439604544569344","6933093663375360","3466546831687680","1422173059153920","474057686384640","126415383035904","26336538132480","4131221667840","459024629760","32212254720","1073741824"],["64268410079232","942603347828736","6598223434801152","29325437488005120","92863885378682880","222873324908838912","420982947050029056","641497824076234752","801872280095293440","831571253432156160","720695086307868672","524141880951177216","320308927247941632","164260988332277760","70397566428119040","25030245841108992","7300488370323456","1717761969487872","318104068423680","44646185041920","4464618504192","283467841536","8589934592"],["1156831381426176","18509302102818816","141904649454944256","693756064001949696","2428146224006823936","6475056597351530496","13669563927742119936","23433538161843634176","33197512395945148416","39345199876675731456","39345199876675731456","33383805955967287296","24110526523754151936","14837247091541016576","7771891333664342016","3454173926073040896","1295315222277390336","406373403067416576","105356067461922816","22180224728825856","3696704121470976","469422745583616","42674795053056","2473901162496","68719476736"],["20822964865671168","360931391004966912","3007761591708057600","16041395155776307200","61492014763809177600","180376576640506920960","420878678827849482240","801673673957808537600","1269316650433196851200","1692422200577595801600","1918078493987941908480","1859954903261034577920","1549962419384195481600","1112793531865576243200","688872186392975769600","367398499409587077120","168390978896060743680","66035677998455193600","22011892666151731200","6178776888744345600","1441714607373680640","274612306166415360","41607925176729600","4824107266867200","402008938905600","21440476741632","549755813888"],["374813367582081024","6996516194865512448","62968645753789612032","363818842133006647296","1515911842220861030400","4850917895106755297280","12396790176383930204160","25974227036232996618240","45454897313407744081920","67340588612455917158400","85298078909110828400640","93052449719029994618880","87882869179083883806720","72109020864889340559360","51506443474920957542400","32048453717728595804160","17359579097102989393920","8169213692754347950080","3328198171122141757440","1167788831972681318400","350336649591804395520","88974387197918576640","18873354860164546560","3282322584376442880","455878136718950400","48627001250021376","3740538557693952","184717953466368","4398046511104"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","50","2500","125000","6250000","312500000","15625000000","781250000000","39062500000000","1953125000000000","97656250000000000","4882812500000000000","244140625000000000000","12207031250000000000000","610351562500000000000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["18","24","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193634,"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-aca210c463f14435","e-e5d62a3bb0d32f6f"]},{"id":193635,"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-aca210c463f14435","e-e5d62a3bb0d32f6f"]},{"id":193636,"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-aca210c463f14435","e-e5d62a3bb0d32f6f"]}],"region_exact":{},"incoming_edges":[{"id":"e-aca210c463f14435","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-af48e57b5ab0efc6","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":"3f24a04c3eadf08213b7843112b17791fbb85feef8747a7ccef72d3005924516","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-4d3a36b3f07ad6a5","role":"sequence","name":"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.","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]"}]},{"id":"e-e5d62a3bb0d32f6f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":2}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-af48e57b5ab0efc6","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"ef77861da28e728847d82588e40d56000f54fd585a6156814540d9e798997c9a","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-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-03eb1b5972a33aa9","kind":"fingerprint_related","node_a_id":"n-1e083728b424e77f","node_b_id":"n-af48e57b5ab0efc6","fingerprint":"f6-6b852c5cd7d4c499","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b852c5cd7d4c499"},"other_id":"n-1e083728b424e77f","other":{"id":"n-1e083728b424e77f","name":"basis_map_A013610 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-6b852c5cd7d4c499","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","12","9"],["16","96","216","216","81"]],"card_rows":[["1"],["4","12","9"],["16","96","216","216","81"],["64","576","2160","4320","4860","2916","729"]],"row_sums_preview":["1","25","625","15625","390625","9765625","244140625","6103515625"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013610"],"edges":["e-cb7f9b19b94bd36e"],"text":"A034870 \u2192 basis A013610"},"headline":"A034870 \u2192 basis A013610","chain":null}},{"id":"l-25bb21e3df3d17cf","kind":"fingerprint_related","node_a_id":"n-af48e57b5ab0efc6","node_b_id":"n-c1a4b2af6b8ff4f9","fingerprint":"f6-6b852c5cd7d4c499","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b852c5cd7d4c499"},"other_id":"n-c1a4b2af6b8ff4f9","other":{"id":"n-c1a4b2af6b8ff4f9","name":"basis_map_A013609 of basis_map_A013610 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":"f6-6b852c5cd7d4c499","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","48","36"],["256","1536","3456","3456","1296"]],"card_rows":[["1"],["16","48","36"],["256","1536","3456","3456","1296"],["4096","36864","138240","276480","311040","186624","46656"]],"row_sums_preview":["1","100","10000","1000000","100000000","10000000000","1000000000000","100000000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["basis A013610","basis A013609"],"edges":["e-4aebfd11077ddb65","e-bfb14b12994a4a5e"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A013610 \u2192 basis A013609"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-29223b85dbf949ce","kind":"fingerprint_related","node_a_id":"n-23c9ebec99e56671","node_b_id":"n-af48e57b5ab0efc6","fingerprint":"f6-6b852c5cd7d4c499","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b852c5cd7d4c499"},"other_id":"n-23c9ebec99e56671","other":{"id":"n-23c9ebec99e56671","name":"shift_x_plus_one 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":"f6-6b852c5cd7d4c499","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","12","4"],["81","216","216","96","16"]],"card_rows":[["1"],["9","12","4"],["81","216","216","96","16"],["729","2916","4860","4320","2160","576","64"]],"row_sums_preview":["1","25","625","15625","390625","9765625","244140625","6103515625"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-20c0b01c4fb345ed","e-606ea858cc2070d2"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-6ff3faf4240ef0e1","kind":"fingerprint_related","node_a_id":"n-87b7b64af43d2638","node_b_id":"n-af48e57b5ab0efc6","fingerprint":"f6-6b852c5cd7d4c499","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b852c5cd7d4c499"},"other_id":"n-87b7b64af43d2638","other":{"id":"n-87b7b64af43d2638","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.","first_index":0,"fingerprint":"f6-6b852c5cd7d4c499","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","24","18"],["64","384","864","864","324"]],"card_rows":[["1"],["8","24","18"],["64","384","864","864","324"],["512","4608","17280","34560","38880","23328","5832"]],"row_sums_preview":["1","50","2500","125000","6250000","312500000","15625000000","781250000000"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["basis A013610"],"edges":["e-a142fb8711dcbce4"],"text":"A139548 \u2192 basis A013610"},"headline":"A139548 \u2192 basis A013610","chain":null}},{"id":"l-c5af6444d3d0f6a2","kind":"fingerprint_related","node_a_id":"n-af48e57b5ab0efc6","node_b_id":"n-08ea8d2f5c08b286","fingerprint":"f6-6b852c5cd7d4c499","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b852c5cd7d4c499"},"other_id":"n-08ea8d2f5c08b286","other":{"id":"n-08ea8d2f5c08b286","name":"basis_map_A013610 of basis_map_A013609 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":"f6-6b852c5cd7d4c499","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["32","96","72"],["1024","6144","13824","13824","5184"]],"card_rows":[["1"],["32","96","72"],["1024","6144","13824","13824","5184"],["32768","294912","1105920","2211840","2488320","1492992","373248"]],"row_sums_preview":["1","200","40000","8000000","1600000000","320000000000","64000000000000","12800000000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["basis A013609","basis A013610"],"edges":["e-65b163cdbb61d2ab","e-d11339927c3a9272"],"text":"A139548 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A139548 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-db12245d4a95de72","kind":"fingerprint_related","node_a_id":"n-fd18f6eb36bb6ac9","node_b_id":"n-af48e57b5ab0efc6","fingerprint":"f6-6b852c5cd7d4c499","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b852c5cd7d4c499"},"other_id":"n-fd18f6eb36bb6ac9","other":{"id":"n-fd18f6eb36bb6ac9","name":"shift_x_plus_one of basis_map_A013611 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-6b852c5cd7d4c499","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["36","48","16"],["1296","3456","3456","1536","256"]],"card_rows":[["1"],["36","48","16"],["1296","3456","3456","1536","256"],["46656","186624","311040","276480","138240","36864","4096"]],"row_sums_preview":["1","100","10000","1000000","100000000","10000000000","1000000000000","100000000000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-a6b5412d61e1c4e3","e-2a7af4bc98279738"],"text":"A034870 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A034870 \u2192 basis A013611 \u2192 x \u21a6 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:           18,            24,              8;\n  2:          324,           864,            864,            384,              64;\n  3:         5832,         23328,          38880,          34560,           17280,            4608,             512;\n  4:       104976,        559872,        1306368,        1741824,         1451520,          774144,          258048,           49152,            4096;\n  5:      1889568,      12597120,       37791360,       67184640,        78382080,        62705664,        34836480,        13271040,         3317760,          491520,           32768;\n  6:     34012224,     272097792,      997691904,     2217093120,      3325639680,      3547348992,      2759049216,      1576599552,       656916480,       194641920,        38928384,        4718592,         262144;\n  7:    612220032,    5714053632,    24760899072,    66029064192,    121053284352,    161404379136,    161404379136,    122974765056,     71735279616,     31882346496,     10627448832,     2576351232,      429391872,      44040192,       2097152;\n  8:  11019960576,  117546246144,   587731230720,  1828497162240,   3961743851520,   6338790162432,   7747410198528,   7378485903360,   5533864427520,   3279327068160,   1530352631808,   556491866112,   154581073920,   31708938240,    4529848320,    402653184,    16777216;\n  9: 198359290368, 2380311484416, 13488431745024, 47958868426752, 119897171066880, 223808052658176, 323278298284032, 369460912324608, 338672502964224, 250868520714240, 150521112428544, 72979933298688, 28381085171712, 8732641591296, 2079200378880, 369635622912, 46204452864, 3623878656, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (18+24*x+8*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[(18 + 24*x + 8*x^2)*P[n - 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] = (18 + 24*x + 8*x^2)*v[i - 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-336459bdce627a87","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-9d7e9cd87d9b797c","e-aca210c463f14435"],"text":"A139548 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A139548 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}