{"id":"n-6be72d7a1f4aa929","revision":0,"first_index":0,"fingerprint":"f6-1bc3b9efcd292160","name":"derivative of basis_map_A013611 of Triangle of coefficients in expansion of (14 + x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[[],["4"],["120","32"],["2700","1440","192"],["54000","43200","11520","1024"],["1012500","1080000","432000","76800","5120"],["18225000","24300000","12960000","3456000","460800","24576"],["318937500","510300000","340200000","120960000","24192000","2580480","114688"],["5467500000","10206000000","8164800000","3628800000","967680000","154828800","13762560","524288"],["92264062500","196830000000","183708000000","97977600000","32659200000","6967296000","928972800","70778880","2359296"],["1537734375000","3690562500000","3936600000000","2449440000000","979776000000","261273600000","46448640000","5308416000","353894400","10485760"],["25372617187500","67660312500000","81192375000000","57736800000000","26943840000000","8622028800000","1916006400000","291962880000","29196288000","1730150400","46137344"],["415188281250000","1217885625000000","1623847500000000","1299078000000000","692841600000000","258660864000000","68976230400000","13138329600000","1751777280000","155713536000","8304721920","201326592"],["6746809570312500","21589790625000000","31665026250000000","28146690000000000","16888014000000000","7205552640000000","2241727488000000","512394854400000","85399142400000","10121379840000","809710387200","39258685440","872415232"],["108986923828125000","377821335937500000","604514137500000000","591080490000000000","394053660000000000","189145756800000000","67251824640000000","17933819904000000","3586763980800000","531372441600000","56679727104000","4122161971200","183207198720","3758096384"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","152","4332","109744","2606420","59426376","1317284668","28603895648","611408269476","12907507911160","269766915343244","5591532427114512","115092375791440372","2354967073886395304"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["15","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190466,"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-25587c881c0f35c7"]},{"id":190467,"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-25587c881c0f35c7"]},{"id":190468,"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-25587c881c0f35c7"]},{"id":190469,"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-25587c881c0f35c7"]}],"region_exact":{},"incoming_edges":[{"id":"e-25587c881c0f35c7","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-6be72d7a1f4aa929","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":"d49259253a0bff789142dcff9811647655867bec1192448aefc3fa6c04d354ec","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-60783defa9dbe6a9","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (14 + x)^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-4f74139b6f3db41b","kind":"fingerprint_related","node_a_id":"n-6be72d7a1f4aa929","node_b_id":"n-440a893d3788ec29","fingerprint":"f6-1bc3b9efcd292160","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1bc3b9efcd292160"},"other_id":"n-440a893d3788ec29","other":{"id":"n-440a893d3788ec29","name":"reversal of basis_map_A013611 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-1bc3b9efcd292160","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","15"],["16","120","225"]],"card_rows":[["1"],["4","15"],["16","120","225"],["64","720","2700","3375"]],"row_sums_preview":["1","19","361","6859","130321","2476099","47045881","893871739"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013611","reverse"],"edges":["e-8d978ffb07d0025e","e-c1b51eb865ae1ef9"],"text":"A147716 \u2192 basis A013611 \u2192 reverse"},"headline":"A147716 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-8c7c2694d4655091","kind":"fingerprint_related","node_a_id":"n-60783defa9dbe6a9","node_b_id":"n-6be72d7a1f4aa929","fingerprint":"f6-1bc3b9efcd292160","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1bc3b9efcd292160"},"other_id":"n-60783defa9dbe6a9","other":{"id":"n-60783defa9dbe6a9","name":"basis_map_A013611 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-1bc3b9efcd292160","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","4"],["225","120","16"]],"card_rows":[["1"],["15","4"],["225","120","16"],["3375","2700","720","64"]],"row_sums_preview":["1","19","361","6859","130321","2476099","47045881","893871739"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013611"],"edges":["e-8d978ffb07d0025e"],"text":"A147716 \u2192 basis A013611"},"headline":"A147716 \u2192 basis A013611","chain":null}},{"id":"l-f8efb902dad6399f","kind":"fingerprint_related","node_a_id":"n-6be72d7a1f4aa929","node_b_id":"n-6cb9a6f2f8e7a17f","fingerprint":"f6-1bc3b9efcd292160","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1bc3b9efcd292160"},"other_id":"n-6cb9a6f2f8e7a17f","other":{"id":"n-6cb9a6f2f8e7a17f","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (1+11x)^n.","first_index":0,"fingerprint":"f6-1bc3b9efcd292160","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","1080","2025"],["1728","19440","72900","91125"],["20736","311040","1749600","4374000","4100625"]],"card_rows":[["144","1080","2025"],["1728","19440","72900","91125"],["20736","311040","1749600","4374000","4100625"],["248832","4665600","34992000","131220000","246037500","184528125"]],"row_sums_preview":["3249","185193","10556001","601692057","34296447249","1954897493193","111429157112001","6351461955384057"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1e393c17b0898c2","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-4217afbeac9f09d9","e-30b085083e32fedd"],"text":"A013618 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A013618 \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:           0;\n  1:           4;\n  2:         120,           32;\n  3:        2700,         1440,          192;\n  4:       54000,        43200,        11520,        1024;\n  5:     1012500,      1080000,       432000,       76800,        5120;\n  6:    18225000,     24300000,     12960000,     3456000,      460800,      24576;\n  7:   318937500,    510300000,    340200000,   120960000,    24192000,    2580480,    114688;\n  8:  5467500000,  10206000000,   8164800000,  3628800000,   967680000,  154828800,  13762560,   524288;\n  9: 92264062500, 196830000000, 183708000000, 97977600000, 32659200000, 6967296000, 928972800, 70778880, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (15*n+4*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((15*n + 4*n*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] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((15*n + 4*n*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-a679863398df5985","steps":["basis A013611","D"],"edges":["e-8d978ffb07d0025e","e-25587c881c0f35c7"],"text":"A147716 \u2192 basis A013611 \u2192 D"},"headline":"A147716 \u2192 basis A013611 \u2192 D","chain":null}