{"id":"n-7ffe75f198ff0f55","revision":0,"first_index":0,"fingerprint":"f6-918ab47be015a3a4","name":"basis_map_A013611 of Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^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"],["3","8"],["21","116","160"],["147","1232","3440","3200"],["1029","11564","48720","91200","64000"],["7203","101528","572320","1612800","2272000","1280000"],["50421","854756","6036800","22736000","48160000","54400000","25600000"],["352947","6991712","59352720","279888000","791840000","1344000000","1267200000","512000000"],["2470629","56000924","555303280","3146270400","11140640000","25244800000","35750400000","28928000000","10240000000"],["17294403","441419048","5007141440","33129958400","140909888000","399526400000","755148800000","917504000000","650240000000","204800000000"],["121060821","3435821396","43878371040","332052537600","1648968384000","5614882560000","13276569600000","21525504000000","22901760000000","14438400000000","4096000000000"],["847425747","26471966192","375865025200","3201935184000","18183829440000","72283545600000","205233638400000","416209920000000","590822400000000","559104000000000","317440000000000","81920000000000"],["5931980229","202252278284","3160494500240","29930846792000","191325509760000","869661408000000","2882306380800000","7018142208000000","12459955200000000","15730176000000000","13404160000000000","6922240000000000","1638400000000000"],["41523861603","1534405552568","26168507067360","272725817548800","1937895504160000","9914140051200000","37569372825600000","106773123072000000","227582530560000000","359310336000000000","408432640000000000","316538880000000000","149913600000000000","32768000000000000"],["290667031221","11571316100036","213867660522880","2432450864188800","19019784880096000","108156890441600000","461268410803200000","1498799318016000000","3728540175360000000","7066822963200000000","10045235200000000000","10384424960000000000","7380172800000000000","3227648000000000000","655360000000000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","297","8019","216513","5845851","157837977","4261625379","115063885233","3106724901291","83881572334857","2264802453041139","61149666232110753","1651040988266990331","44578106683208738937"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","8"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-7","-20"],["7","20"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":205594,"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-9453ba0e1b53d61a"]},{"id":205595,"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-9453ba0e1b53d61a"]},{"id":205596,"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-9453ba0e1b53d61a"]},{"id":205597,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-9453ba0e1b53d61a"]}],"region_exact":{},"incoming_edges":[{"id":"e-9453ba0e1b53d61a","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-7ffe75f198ff0f55","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":"7eeb2df9bebe2cca1a87129abcac1350e153b5a7870b59b1b397b5db37dcd48d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-cedb1276760d9bb1","role":"sequence","name":"Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^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":[],"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:        3,         8;\n  2:       21,       116,        160;\n  3:      147,      1232,       3440,        3200;\n  4:     1029,     11564,      48720,       91200,        64000;\n  5:     7203,    101528,     572320,     1612800,      2272000,      1280000;\n  6:    50421,    854756,    6036800,    22736000,     48160000,     54400000,     25600000;\n  7:   352947,   6991712,   59352720,   279888000,    791840000,   1344000000,   1267200000,    512000000;\n  8:  2470629,  56000924,  555303280,  3146270400,  11140640000,  25244800000,  35750400000,  28928000000,  10240000000;\n  9: 17294403, 441419048, 5007141440, 33129958400, 140909888000, 399526400000, 755148800000, 917504000000, 650240000000, 204800000000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(20*n-20)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (20*n - 20)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (20*n - 20)*x)*v[i - 1])/(n - 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-cedb1276760d9bb1","steps":["basis A013611"],"edges":["e-9453ba0e1b53d61a"],"text":"A193726 \u2192 basis A013611"},"headline":"A193726 \u2192 basis A013611","chain":null}
