{"id":"n-245f1aee5991880d","revision":0,"first_index":0,"fingerprint":"f6-49793563bcf540fa","name":"basis_map_A013609 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","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":[["625","1200","576"],["15625","45000","43200","13824"],["390625","1500000","2160000","1382400","331776"],["9765625","46875000","90000000","86400000","41472000","7962624"],["244140625","1406250000","3375000000","4320000000","3110400000","1194393600","191102976"],["6103515625","41015625000","118125000000","189000000000","181440000000","104509440000","33443020800","4586471424"],["152587890625","1171875000000","3937500000000","7560000000000","9072000000000","6967296000000","3344302080000","917294284800","110075314176"],["3814697265625","32958984375000","126562500000000","283500000000000","408240000000000","391910400000000","250822656000000","103195607040000","24766945689600","2641807540224"],["95367431640625","915527343750000","3955078125000000","10125000000000000","17010000000000000","19595520000000000","15676416000000000","8599633920000000","3095868211200000","660451885056000","63403380965376"],["2384185791015625","25177001953125000","120849609375000000","348046875000000000","668250000000000000","898128000000000000","862202880000000000","591224832000000000","283787919360000000","90812134195200000","17435929765478400","1521681143169024"],["59604644775390625","686645507812500000","3625488281250000000","11601562500000000000","25059375000000000000","38491200000000000000","43110144000000000000","35473489920000000000","21284093952000000000","9081213419520000000","2615389464821760000","456504342950707200","36520347436056576"],["1490116119384765625","18596649169921875000","107116699218750000000","377050781250000000000","904921875000000000000","1563705000000000000000","2001542400000000000000","1921480704000000000000","1383466106880000000000","737848590336000000000","283333858689024000000","74181955729489920000","11869112916718387200","876488338465357824"],["37252902984619140625","500679016113281250000","3124237060546875000000","11997070312500000000000","31672265625000000000000","60810750000000000000000","87567480000000000000000","96074035200000000000000","80702189568000000000000","51649401323520000000000","24791712635289600000000","8654561501773824000000","2077094760425717760000","306770918462875238400","21035720123168587776"],["931322574615478515625","13411045074462890625000","90122222900390625000000","374908447265625000000000","1079736328125000000000000","2280403125000000000000000","3648645000000000000000000","4503470400000000000000000","4323331584000000000000000","3228087582720000000000000","1859378447646720000000000","811365140791296000000000","259636845053214720000000","57519547211789107200000","7888395046188220416000","504857282956046106624"],["23283064365386962890625","357627868652343750000000","2574920654296875000000000","11535644531250000000000000","35991210937500000000000000","82923750000000000000000000","145945800000000000000000000","200154240000000000000000000","216166579200000000000000000","184462147584000000000000000","123958563176448000000000000","64909211263303680000000000","25963684505321472000000000","7669272961571880960000000","1577679009237644083200000","201942913182418442649600","12116574790945106558976"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2401","117649","5764801","282475249","13841287201","678223072849","33232930569601","1628413597910449","79792266297612001","3909821048582988049","191581231380566414401","9387480337647754305649","459986536544739960976801","22539340290692258087863249","1104427674243920646305299201"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["625","1200","576"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["25","24"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201115,"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-5885d471be6c0dbe"]},{"id":201116,"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-5885d471be6c0dbe"]},{"id":201117,"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-5885d471be6c0dbe"]}],"region_exact":{},"incoming_edges":[{"id":"e-5885d471be6c0dbe","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-245f1aee5991880d","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":"cd042eb365fa0231f9b36bcd4dcbe6f16a10dd773a4cf26f019db54ccffd976a","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-b8a3f88eab200094","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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]"}]}],"used_by":[],"used_by_total":0,"links":[],"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:              625,              1200,                576;\n  1:            15625,             45000,              43200,              13824;\n  2:           390625,           1500000,            2160000,            1382400,             331776;\n  3:          9765625,          46875000,           90000000,           86400000,           41472000,            7962624;\n  4:        244140625,        1406250000,         3375000000,         4320000000,         3110400000,         1194393600,          191102976;\n  5:       6103515625,       41015625000,       118125000000,       189000000000,       181440000000,       104509440000,        33443020800,         4586471424;\n  6:     152587890625,     1171875000000,      3937500000000,      7560000000000,      9072000000000,      6967296000000,      3344302080000,       917294284800,       110075314176;\n  7:    3814697265625,    32958984375000,    126562500000000,    283500000000000,    408240000000000,    391910400000000,    250822656000000,    103195607040000,     24766945689600,     2641807540224;\n  8:   95367431640625,   915527343750000,   3955078125000000,  10125000000000000,  17010000000000000,  19595520000000000,  15676416000000000,   8599633920000000,   3095868211200000,   660451885056000,    63403380965376;\n  9: 2384185791015625, 25177001953125000, 120849609375000000, 348046875000000000, 668250000000000000, 898128000000000000, 862202880000000000, 591224832000000000, 283787919360000000, 90812134195200000, 17435929765478400, 1521681143169024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (25+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 625 + 1200*x + 576*x^2.","mathematica":"P[0, x_] := 625 + 1200*x + 576*x^2;\nP[n_, x_] := P[n, x] = Expand[(25 + 24*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] = 625 + 1200*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (25 + 24*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-4397e742863fdbe2","steps":["basis A013611","basis A013609"],"edges":["e-7b93be005ec5b750","e-5885d471be6c0dbe"],"text":"A038305 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A038305 \u2192 basis A013611 \u2192 basis A013609","chain":null}
