{"id":"n-7dc80baa1678cc50","revision":0,"first_index":0,"fingerprint":"f6-87f28ed98a2aba0a","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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":[["160","168","36"],["1664","2880","1512","216"],["16384","39936","34560","12096","1296"],["155648","491520","599040","345600","90720","7776"],["1441792","5603328","8847360","7188480","3110400","653184","46656"],["13107200","60555264","117669888","123863040","75479040","26127360","4572288","279936"],["117440512","629145600","1453326336","1882718208","1486356480","724598784","209018880","31352832","1679616"],["1040187392","6341787648","16986931200","26159874048","25416695808","16052649984","6521389056","1612431360","211631616","10077696"],["9126805504","62411243520","190253629440","339738624000","392398110720","305000349696","160526499840","55897620480","12093235200","1410877440","60466176"],["79456894976","602369163264","2059571036160","4185579847680","5605687296000","5179655061504","3355003846656","1513535569920","461155368960","88683724800","9311791104","362797056"],["687194767360","5720896438272","21685289877504","49429704867840","75340437258240","80721897062400","62155860738048","34508610994176","13621820129280","3689242951680","638522818560","60949905408","2176782336"],["5909874999296","53601191854080","223114961092608","563817536815104","963879244922880","1175310821228544","1049384661811200","692593876795392","336458957193216","118055774453760","28776095023104","4527707258880","396174385152","13060694016"],["50577534877696","496429499940864","2251250057871360","6247218910593024","11840168273117184","16193171314704384","16454351497199616","12592615941734400","7272235706351616","3140283600470016","991668505411584","219744725630976","31693950812160","2559896027136","78364164096"],["431008558088192","4551978138992640","22339327497338880","67537501736140800","140562425488343040","213123028916109312","242897569720565760","211555947821137920","141666929344512000","72722357063516160","28262552404230144","8113651407912960","1648085442232320","219419659468800","16456474460160","470184984576"],["3659174697238528","41376821576466432","218494950671646720","714858479914844160","1620900041667379200","2698798569376186368","3409968462657748992","3331166670453473280","2538671373853655040","1511113913008128000","698134627809755136","246655002800553984","64909211263303680","12170477111869440","1504591950643200","105321436545024","2821109907456"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["364","6272","104272","1690304","26891200","421654016","6535637248","100353655808","1528917462016","23140372398080","348262604591104","5215839938527232","77784047778906112","1155648709858033664","17112490511359344640"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["160","168","36"]],"recurrence":{"denominator":{"coeffs":[["7","3"],["3","0"]]},"terms":[{"coeff":{"coeffs":[["80","84","18"],["24","18","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192067,"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-36faf349c7bd0aed"]},{"id":192068,"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-36faf349c7bd0aed"]},{"id":192069,"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-36faf349c7bd0aed"]},{"id":192070,"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-36faf349c7bd0aed"]}],"region_exact":{},"incoming_edges":[{"id":"e-36faf349c7bd0aed","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-7dc80baa1678cc50","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"94954302053027d1e00d152bec506237929b089605e51ce34580d9fe69014c60","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c11cb6bca0e3e7dc","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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":[{"id":"l-4670b257576823bb","kind":"fingerprint_related","node_a_id":"n-7dc80baa1678cc50","node_b_id":"n-8c5ad656fba85f85","fingerprint":"f6-87f28ed98a2aba0a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-87f28ed98a2aba0a"},"other_id":"n-8c5ad656fba85f85","other":{"id":"n-8c5ad656fba85f85","name":"one_plus_derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-87f28ed98a2aba0a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["640","672","144"],["13312","23040","12096","1728"],["262144","638976","552960","193536","20736"]],"card_rows":[["640","672","144"],["13312","23040","12096","1728"],["262144","638976","552960","193536","20736"],["4980736","15728640","19169280","11059200","2903040","248832"]],"row_sums_preview":["1456","50176","1668352","54089728","1721036800","53971714048","1673123135488","51381071773696"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013610","D+1"],"edges":["e-77aaa200c8a95097","e-225d37c726225e58"],"text":"A038330 \u2192 basis A013610 \u2192 D+1"},"headline":"A038330 \u2192 basis A013610 \u2192 D+1","chain":null}},{"id":"l-5c0872cb1f3b443f","kind":"fingerprint_related","node_a_id":"n-e63fbd290efa0220","node_b_id":"n-7dc80baa1678cc50","fingerprint":"f6-87f28ed98a2aba0a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-87f28ed98a2aba0a"},"other_id":"n-e63fbd290efa0220","other":{"id":"n-e63fbd290efa0220","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-87f28ed98a2aba0a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["21","9"],["360","378","81"]],"card_rows":[["1"],["21","9"],["360","378","81"],["5616","9720","5103","729"]],"row_sums_preview":["1","30","819","21168","527877","12835746","306307575","7204354164"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["D+1"],"edges":["e-8f0ce1d5978cf162"],"text":"A038335 \u2192 D+1"},"headline":"A038335 \u2192 D+1","chain":null}},{"id":"l-f280faa8e2642ad8","kind":"fingerprint_related","node_a_id":"n-4bf16590d3e16b9c","node_b_id":"n-7dc80baa1678cc50","fingerprint":"f6-87f28ed98a2aba0a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-87f28ed98a2aba0a"},"other_id":"n-4bf16590d3e16b9c","other":{"id":"n-4bf16590d3e16b9c","name":"one_plus_derivative of basis_map_A007318 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-87f28ed98a2aba0a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","3"],["40","42","9"]],"card_rows":[["1"],["7","3"],["40","42","9"],["208","360","189","27"]],"row_sums_preview":["1","10","91","784","6517","52822","420175","3294172"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["basis A007318","D+1"],"edges":["e-5bade392cbb9f615","e-8a776f8940c815b2"],"text":"A013610 \u2192 basis A007318 \u2192 D+1"},"headline":"A013610 \u2192 basis A007318 \u2192 D+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:         160,          168,            36;\n  1:        1664,         2880,          1512,           216;\n  2:       16384,        39936,         34560,         12096,          1296;\n  3:      155648,       491520,        599040,        345600,         90720,          7776;\n  4:     1441792,      5603328,       8847360,       7188480,       3110400,        653184,         46656;\n  5:    13107200,     60555264,     117669888,     123863040,      75479040,      26127360,       4572288,        279936;\n  6:   117440512,    629145600,    1453326336,    1882718208,    1486356480,     724598784,     209018880,      31352832,      1679616;\n  7:  1040187392,   6341787648,   16986931200,   26159874048,   25416695808,   16052649984,    6521389056,    1612431360,    211631616,    10077696;\n  8:  9126805504,  62411243520,  190253629440,  339738624000,  392398110720,  305000349696,  160526499840,   55897620480,  12093235200,  1410877440,   60466176;\n  9: 79456894976, 602369163264, 2059571036160, 4185579847680, 5605687296000, 5179655061504, 3355003846656, 1513535569920, 461155368960, 88683724800, 9311791104, 362797056;\n","recurrence_sentence":"The row polynomials satisfy (3*n+7+3*x)*P(n,x) = (24*n+80+(18*n+84)*x+18*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 160 + 168*x + 36*x^2.","mathematica":"P[0, x_] := 160 + 168*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((24*n + 80 + (18*n + 84)*x + 18*x^2)*P[n - 1, x])/(3*n + 7 + 3*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 160 + 168*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24*n + 80 + (18*n + 84)*x + 18*x^2)*v[i - 1])/(3*n + 7 + 3*x)); 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-c11cb6bca0e3e7dc","steps":["D+1"],"edges":["e-36faf349c7bd0aed"],"text":"A038284 \u2192 D+1"},"headline":"A038284 \u2192 D+1","chain":null}