{"id":"n-0176d4aaeb293b4e","revision":0,"first_index":0,"fingerprint":"f6-de8557183b81eda9","name":"shift_x_plus_one of basis_map_A013611 of Triangle of coefficients in expansion of (1+5x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["676","1040","400"],["17576","40560","31200","8000"],["456976","1406080","1622400","832000","160000"],["11881376","45697600","70304000","54080000","20800000","3200000"],["308915776","1425765120","2741856000","2812160000","1622400000","499200000","64000000"],["8031810176","43248208640","99803558400","127953280000","98425600000","45427200000","11648000000","1280000000"],["208827064576","1285089628160","3459856691200","5322856448000","5118131200000","3149619200000","1211392000000","266240000000","25600000000"],["5429503678976","37588871623680","115658066534400","207591401472000","239528540160000","184252723200000","94488576000000","31150080000000","5990400000000","512000000000"],["141167095653376","1085900735795200","3758887162368000","7710537768960000","10379570073600000","9581141606400000","6141757440000000","2699673600000000","778752000000000","133120000000000","10240000000000"],["3670344486987776","31056761043742720","119449080937472000","275651725240320000","424079577292800000","456701083238400000","351308525568000000","193026662400000000","74241024000000000","19036160000000000","2928640000000000","204800000000000"],["95428956661682176","880882676877066240","3726811325249126400","9555926474997760000","16539103514419200000","20355819710054400000","18268043329536000000","12044863733760000000","5790799872000000000","1979760640000000000","456867840000000000","63897600000000000","4096000000000000"],["2481152873203736576","24811528732037365760","114514747994018611200","322990314854924288000","621135220874854400000","860033382749798400000","882085520769024000000","678527323668480000000","391458071347200000000","167289774080000000000","51473776640000000000","10798694400000000000","1384448000000000000","81920000000000000"],["64509974703297150976","694722804497046241280","3473614022485231206400","10688043146108403712000","22609322039844700160000","34783572368991846400000","40134891194990592000000","35283420830760960000000","23748456328396800000000","12178695553024000000000","4684113674240000000000","1310241587200000000000","251969536000000000000","29818880000000000000","1638400000000000000"],["1677259342285725925376","19352992410989145292800","104208420674556936192000","347361402248523120640000","801603235958130278400000","1356559322390682009600000","1739178618449592320000000","1720066765499596800000000","1323128281153536000000000","791615210946560000000000","365360866590720000000000","127748554752000000000000","32756039680000000000000","5814681600000000000000","638976000000000000000","32768000000000000000"],["43608742899428874059776","536722989531432296120320","3096478785758263246848000","11115564871952739860480000","27788912179881849651200000","51302607101320337817600000","72349830527503040512000000","79505308271981363200000000","68802670619983872000000000","47044561107681280000000000","25331686750289920000000000","10628679755366400000000000","3406628126720000000000000","806302515200000000000000","132907008000000000000000","13631488000000000000000","655360000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056","42420747482776576","1951354384207722496","89762301673555234816","4129065876983540801536","189937030341242876870656","8737103395697172336050176","401906756202069927458308096"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["676","1040","400"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["26","20"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194206,"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-0e175e2e6f6bc81f"]},{"id":194207,"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-0464495adf650367","e-0a9e45dc5aea6609","e-0e175e2e6f6bc81f"]},{"id":194208,"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-0464495adf650367","e-0a9e45dc5aea6609","e-0e175e2e6f6bc81f"]},{"id":194209,"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-0e175e2e6f6bc81f"]},{"id":194210,"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-0464495adf650367","e-0a9e45dc5aea6609"]}],"region_exact":{},"incoming_edges":[{"id":"e-0464495adf650367","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-0176d4aaeb293b4e","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":"0a0c1d5c4801b378404261fe29916c56719d7b8105983e9eaf38eb1f902844e1","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-a2769223501d377d","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+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-0a9e45dc5aea6609","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-0176d4aaeb293b4e","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":"1e77b13615fac57534ce80cf13a1e341456e0e5b39053c448fadf944511dcf24","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-cf1de4f27da354f5","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","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-0e175e2e6f6bc81f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-0176d4aaeb293b4e","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":"c49cad41a3c70821c1893f1b5fcf50f77aec116f464f5fa62dc556ee31841beb","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-a2769223501d377d","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+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-2880cbc0af593f00","kind":"fingerprint_related","node_a_id":"n-0176d4aaeb293b4e","node_b_id":"n-488f274140470236","fingerprint":"f6-de8557183b81eda9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-de8557183b81eda9"},"other_id":"n-488f274140470236","other":{"id":"n-488f274140470236","name":"basis_map_A013609 of basis_map_A007318 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-de8557183b81eda9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["13","10"],["169","260","100"]],"card_rows":[["1"],["13","10"],["169","260","100"],["2197","5070","3900","1000"]],"row_sums_preview":["1","23","529","12167","279841","6436343","148035889","3404825447"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A007318","basis A013609"],"edges":["e-31008ec5d02c6da5","e-5034b254c9504d73"],"text":"A013622 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A013622 \u2192 basis A007318 \u2192 basis A013609","chain":null}},{"id":"l-32b85dc09d46dd18","kind":"fingerprint_related","node_a_id":"n-0541c88feea53d15","node_b_id":"n-0176d4aaeb293b4e","fingerprint":"f6-de8557183b81eda9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-de8557183b81eda9"},"other_id":"n-0541c88feea53d15","other":{"id":"n-0541c88feea53d15","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-de8557183b81eda9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["100","260","169"],["1000","3900","5070","2197"],["10000","52000","101400","87880","28561"]],"card_rows":[["100","260","169"],["1000","3900","5070","2197"],["10000","52000","101400","87880","28561"],["100000","650000","1690000","2197000","1428050","371293"]],"row_sums_preview":["529","12167","279841","6436343","148035889","3404825447","78310985281","1801152661463"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["x \u21a6 x/(x+1)"],"edges":["e-256c5bd1a0f586bb"],"text":"A038305 \u2192 x \u21a6 x/(x+1)"},"headline":"A038305 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-a2ad23716f3c099a","kind":"fingerprint_related","node_a_id":"n-e3018af0169eeeab","node_b_id":"n-0176d4aaeb293b4e","fingerprint":"f6-de8557183b81eda9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-de8557183b81eda9"},"other_id":"n-e3018af0169eeeab","other":{"id":"n-e3018af0169eeeab","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-de8557183b81eda9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","13"],["100","260","169"]],"card_rows":[["1"],["10","13"],["100","260","169"],["1000","3900","5070","2197"]],"row_sums_preview":["1","23","529","12167","279841","6436343","148035889","3404825447"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-8d10edd099560dc3","e-235c2ff3553912c8"],"text":"A038291 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A038291 \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:              676,              1040,                400;\n  1:            17576,             40560,              31200,               8000;\n  2:           456976,           1406080,            1622400,             832000,             160000;\n  3:         11881376,          45697600,           70304000,           54080000,           20800000,            3200000;\n  4:        308915776,        1425765120,         2741856000,         2812160000,         1622400000,          499200000,           64000000;\n  5:       8031810176,       43248208640,        99803558400,       127953280000,        98425600000,        45427200000,        11648000000,         1280000000;\n  6:     208827064576,     1285089628160,      3459856691200,      5322856448000,      5118131200000,      3149619200000,      1211392000000,       266240000000,       25600000000;\n  7:    5429503678976,    37588871623680,    115658066534400,    207591401472000,    239528540160000,    184252723200000,     94488576000000,     31150080000000,     5990400000000,      512000000000;\n  8:  141167095653376,  1085900735795200,   3758887162368000,   7710537768960000,  10379570073600000,   9581141606400000,   6141757440000000,   2699673600000000,   778752000000000,   133120000000000,   10240000000000;\n  9: 3670344486987776, 31056761043742720, 119449080937472000, 275651725240320000, 424079577292800000, 456701083238400000, 351308525568000000, 193026662400000000, 74241024000000000, 19036160000000000, 2928640000000000, 204800000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (26+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 676 + 1040*x + 400*x^2.","mathematica":"P[0, x_] := 676 + 1040*x + 400*x^2;\nP[n_, x_] := P[n, x] = Expand[(26 + 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] = 676 + 1040*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (26 + 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-bbfc5a54ff13a785","steps":["basis A013611","basis A007318"],"edges":["e-db3287eb465d1cf9","e-0464495adf650367"],"text":"A013612 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A013612 \u2192 basis A013611 \u2192 basis A007318","chain":null}
