{"id":"n-eeb76685a977fceb","revision":0,"first_index":0,"fingerprint":"f6-f96e0317bce1f626","name":"shift_x_plus_one of basis_map_A013611 of Triangle of coefficients in expansion of (3+5x)^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"],["28","20"],["784","1120","400"],["21952","47040","33600","8000"],["614656","1756160","1881600","896000","160000"],["17210368","61465600","87808000","62720000","22400000","3200000"],["481890304","2065244160","3687936000","3512320000","1881600000","537600000","64000000"],["13492928512","67464642560","144567091200","172103680000","122931200000","52684800000","12544000000","1280000000"],["377801998336","2158868561920","5397171404800","7710244864000","6884147200000","3933798400000","1404928000000","286720000000","25600000000"],["10578455953408","68004359700480","194298170572800","323830284288000","346961018880000","247829299200000","118013952000000","36126720000000","6451200000000","512000000000"],["296196766695424","2115691190681600","6800435970048000","12953211371520000","16191514214400000","13878440755200000","8260976640000000","3371827200000000","903168000000000","143360000000000","10240000000000"],["8293509467471872","65163288672993280","232726030974976000","498698637803520000","712426625433600000","712426625433600000","508876161024000000","259630694400000000","92725248000000000","22077440000000000","3153920000000000","204800000000000"],["232218265089212416","1990442272193249280","7819594640759193600","18618082477998080000","29921918268211200000","34196478020812800000","28497065017344000000","17447182663680000000","7788920832000000000","2472673280000000000","529858560000000000","68812800000000000","4096000000000000"],["6502111422497947648","60376748923195228160","258757495385122406400","677698202199130112000","1210175361069875200000","1555939749946982400000","1481847380901888000000","1058462414929920000000","567033436569600000000","225013268480000000000","64289505280000000000","12523929600000000000","1490944000000000000","81920000000000000"],["182059119829942534144","1820591198299425341440","8452744849247331942400","24150699569278091264000","47438874153939107840000","67769820219913011200000","72610521664192512000000","59273895236075520000000","37046184522547200000000","17641040248832000000000","6300371517440000000000","1636460134400000000000","292225024000000000000","32112640000000000000","1638400000000000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","48","2304","110592","5308416","254803968","12230590464","587068342272","28179280429056","1352605460594688","64925062108545024","3116402981210161152","149587343098087735296","7180192468708211294208","344649238497994142121984"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["28","20"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192030,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-62d85a500dc72ecc"]},{"id":192031,"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-244f54e1bf9ae19d","e-62d85a500dc72ecc"]},{"id":192032,"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-244f54e1bf9ae19d","e-62d85a500dc72ecc"]},{"id":192033,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-62d85a500dc72ecc"]},{"id":192034,"position":4,"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-244f54e1bf9ae19d"]}],"region_exact":{},"incoming_edges":[{"id":"e-244f54e1bf9ae19d","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-eeb76685a977fceb","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":"27be8d29b4fc7a2d2bc352b04844dd18faf1cba8a1d7e2122ebcce14f3e63f85","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-815b996b7a3cb30d","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (3+5x)^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]"}]},{"id":"e-62d85a500dc72ecc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-eeb76685a977fceb","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"9cae7a506b7d7e770a31fd3b547ee5f9ff94e218ef8ef922f136d8562ba2c045","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-815b996b7a3cb30d","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (3+5x)^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-2a0c9b6ab8237922","kind":"fingerprint_related","node_a_id":"n-bb9d1d7a965ec78a","node_b_id":"n-eeb76685a977fceb","fingerprint":"f6-f96e0317bce1f626","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f96e0317bce1f626"},"other_id":"n-bb9d1d7a965ec78a","other":{"id":"n-bb9d1d7a965ec78a","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","first_index":0,"fingerprint":"f6-f96e0317bce1f626","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","7"],["25","70","49"]],"card_rows":[["1"],["5","7"],["25","70","49"],["125","525","735","343"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-7f6faf86ff30c5bb","e-dc2a52545022505a"],"text":"A038220 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A038220 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-336ed2b0fc82ef39","kind":"fingerprint_related","node_a_id":"n-eeb76685a977fceb","node_b_id":"n-1d9325b8d0810008","fingerprint":"f6-f96e0317bce1f626","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f96e0317bce1f626"},"other_id":"n-1d9325b8d0810008","other":{"id":"n-1d9325b8d0810008","name":"shift_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-f96e0317bce1f626","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["441","630","225"],["9261","19845","14175","3375"],["194481","555660","595350","283500","50625"]],"card_rows":[["441","630","225"],["9261","19845","14175","3375"],["194481","555660","595350","283500","50625"],["4084101","14586075","20837250","14883750","5315625","759375"]],"row_sums_preview":["1296","46656","1679616","60466176","2176782336","78364164096","2821109907456","101559956668416"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013610","basis A007318"],"edges":["e-46de8f4df4fd1cb3","e-39d66391a6d06c77"],"text":"A013612 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A013612 \u2192 basis A013610 \u2192 basis A007318","chain":null}},{"id":"l-3cc9f9da1d907c7a","kind":"fingerprint_related","node_a_id":"n-eeb76685a977fceb","node_b_id":"n-dd8520417c7f5dfb","fingerprint":"f6-f96e0317bce1f626","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f96e0317bce1f626"},"other_id":"n-dd8520417c7f5dfb","other":{"id":"n-dd8520417c7f5dfb","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-f96e0317bce1f626","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","1120","784"],["8000","33600","47040","21952"],["160000","896000","1881600","1756160","614656"]],"card_rows":[["400","1120","784"],["8000","33600","47040","21952"],["160000","896000","1881600","1756160","614656"],["3200000","22400000","62720000","87808000","61465600","17210368"]],"row_sums_preview":["2304","110592","5308416","254803968","12230590464","587068342272","28179280429056","1352605460594688"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-4a3b4bf6a80699a3","e-89f156d6b8de7a8e"],"text":"A038334 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A038334 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-40966523589b1503","kind":"fingerprint_related","node_a_id":"n-6a2532ecdfb2c281","node_b_id":"n-eeb76685a977fceb","fingerprint":"f6-f96e0317bce1f626","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f96e0317bce1f626"},"other_id":"n-6a2532ecdfb2c281","other":{"id":"n-6a2532ecdfb2c281","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-f96e0317bce1f626","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","14"],["100","280","196"]],"card_rows":[["1"],["10","14"],["100","280","196"],["1000","4200","5880","2744"]],"row_sums_preview":["1","24","576","13824","331776","7962624","191102976","4586471424"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-8c5f302cdd82df95","e-350229d1d8eb03ef"],"text":"A038291 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A038291 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-55a6d635c663935c","kind":"fingerprint_related","node_a_id":"n-a6a633b989da1879","node_b_id":"n-eeb76685a977fceb","fingerprint":"f6-f96e0317bce1f626","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f96e0317bce1f626"},"other_id":"n-a6a633b989da1879","other":{"id":"n-a6a633b989da1879","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-f96e0317bce1f626","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","5"],["49","70","25"]],"card_rows":[["1"],["7","5"],["49","70","25"],["343","735","525","125"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A038271"],"derivation":{"root":"n-a6a633b989da1879","steps":[],"edges":[],"text":"A038271"},"headline":"A038271: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","chain":null}},{"id":"l-6e3a489c59d62aee","kind":"fingerprint_related","node_a_id":"n-eeb76685a977fceb","node_b_id":"n-d49e9ad0468aee45","fingerprint":"f6-f96e0317bce1f626","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f96e0317bce1f626"},"other_id":"n-d49e9ad0468aee45","other":{"id":"n-d49e9ad0468aee45","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-f96e0317bce1f626","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","630","441"],["3375","14175","19845","9261"],["50625","283500","595350","555660","194481"]],"card_rows":[["225","630","441"],["3375","14175","19845","9261"],["50625","283500","595350","555660","194481"],["759375","5315625","14883750","20837250","14586075","4084101"]],"row_sums_preview":["1296","46656","1679616","60466176","2176782336","78364164096","2821109907456","101559956668416"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-f3e6c91e6b1bdc2d","e-bb623df4f2e14ef0"],"text":"A038329 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038329 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-927d31dea3551ab6","kind":"fingerprint_related","node_a_id":"n-d0305c3b60911dde","node_b_id":"n-eeb76685a977fceb","fingerprint":"f6-f96e0317bce1f626","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f96e0317bce1f626"},"other_id":"n-d0305c3b60911dde","other":{"id":"n-d0305c3b60911dde","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-f96e0317bce1f626","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["5"],["70","50"]],"card_rows":[["5"],["70","50"],["735","1050","375"],["6860","14700","10500","2500"]],"row_sums_preview":["0","5","120","2160","34560","518400","7464960","104509440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["D"],"edges":["e-4746065a5945cdac"],"text":"A038271 \u2192 D"},"headline":"A038271 \u2192 D","chain":null}},{"id":"l-c3e1dd527443a127","kind":"fingerprint_related","node_a_id":"n-ba58052bba61b079","node_b_id":"n-eeb76685a977fceb","fingerprint":"f6-f96e0317bce1f626","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f96e0317bce1f626"},"other_id":"n-ba58052bba61b079","other":{"id":"n-ba58052bba61b079","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-f96e0317bce1f626","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","7"],["25","70","49"]],"card_rows":[["1"],["5","7"],["25","70","49"],["125","525","735","343"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["reverse"],"edges":["e-e1b8266ed3ef8b8a"],"text":"A038271 \u2192 reverse"},"headline":"A038271 \u2192 reverse","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:             28,             20;\n  2:            784,           1120,             400;\n  3:          21952,          47040,           33600,            8000;\n  4:         614656,        1756160,         1881600,          896000,          160000;\n  5:       17210368,       61465600,        87808000,        62720000,        22400000,         3200000;\n  6:      481890304,     2065244160,      3687936000,      3512320000,      1881600000,       537600000,        64000000;\n  7:    13492928512,    67464642560,    144567091200,    172103680000,    122931200000,     52684800000,     12544000000,     1280000000;\n  8:   377801998336,  2158868561920,   5397171404800,   7710244864000,   6884147200000,   3933798400000,   1404928000000,   286720000000,   25600000000;\n  9: 10578455953408, 68004359700480, 194298170572800, 323830284288000, 346961018880000, 247829299200000, 118013952000000, 36126720000000, 6451200000000, 512000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (28+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(28 + 20*x)*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] = (28 + 20*x)*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-c44f7114310405ad","steps":["basis A013611","basis A007318"],"edges":["e-96f2019a633e1196","e-244f54e1bf9ae19d"],"text":"A013622 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A013622 \u2192 basis A013611 \u2192 basis A007318","chain":null}
