{"id":"n-2ae0099201780429","revision":0,"first_index":0,"fingerprint":"f6-951a7e0f7b8ffcff","name":"one_plus_derivative of basis_map_A013609 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":[["156","320","100"],["1296","4680","4800","1000"],["9936","51840","93600","64000","10000"],["72576","496800","1296000","1560000","800000","100000"],["513216","4354560","14904000","25920000","23400000","9600000","1000000"],["3545856","35925120","152409600","347760000","453600000","327600000","112000000","10000000"],["24074496","283668480","1437004800","4064256000","6955200000","7257600000","4368000000","1280000000","100000000"],["161243136","2166704640","12765081600","43110144000","91445760000","125193600000","108864000000","56160000000","14400000000","1000000000"],["1068235776","16124313600","108335232000","425502720000","1077753600000","1828915200000","2086560000000","1555200000000","702000000000","160000000000","10000000000"],["7014076416","117505935360","886837248000","3972291840000","11701324800000","23710579200000","33530112000000","32788800000000","21384000000000","8580000000000","1760000000000","100000000000"],["45712429056","841689169920","7050356121600","35473489920000","119168755200000","280831795200000","474211584000000","574801920000000","491832000000000","285120000000000","102960000000000","19200000000000","1000000000000"],["296042397696","5942615777280","54709796044800","305515431936000","1152888422400000","3098387635200000","6084688896000000","8806786560000000","9340531200000000","7104240000000000","3706560000000000","1216800000000000","208000000000000","10000000000000"],["1906861326336","41445935677440","415983104409600","2553123815424000","10693040117760000","32280875827200000","72295711488000000","121693777920000000","154118764800000000","145297152000000000","99459360000000000","47174400000000000","14196000000000000","2240000000000000","100000000000000"],["12224809598976","286029198950400","3108445175808000","20799155220480000","95742143078400000","320791203532800000","807021895680000000","1549193817600000000","2281758336000000000","2568646080000000000","2179457280000000000","1356264000000000000","589680000000000000","163800000000000000","24000000000000000","1000000000000000"],["78050707439616","1955969535836160","22882335916032000","165783742709760000","831966208819200000","3063748578508800000","8554432094208000000","18446214758400000000","30983876352000000000","40564592640000000000","41098337280000000000","31701196800000000000","18083520000000000000","7257600000000000000","1872000000000000000","256000000000000000","10000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["576","11776","229376","4325376","79691776","1442840576","25769803776","455266533376","7971459301376","138538465099776","2392537302040576","41095346599755776","702561541869797376","11961560610296037376","202914184810805067776"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["156","320","100"]],"recurrence":{"denominator":{"coeffs":[["8","5"],["5","0"]]},"terms":[{"coeff":{"coeffs":[["78","160","50"],["30","50","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192431,"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-e1d4711e5a448783"]},{"id":192432,"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-e1d4711e5a448783"]},{"id":192433,"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-e1d4711e5a448783"]},{"id":192434,"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-e1d4711e5a448783"]}],"region_exact":{},"incoming_edges":[{"id":"e-e1d4711e5a448783","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-2ae0099201780429","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":"2d72df8f9551be4880b30de5f9f7288f336bafbaceb3b3696865a2a5987c44b4","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-d524cbe046c3619e","role":"sequence","name":"basis_map_A013609 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-35337b10cd419f17","kind":"fingerprint_related","node_a_id":"n-b99042e296c8e644","node_b_id":"n-2ae0099201780429","fingerprint":"f6-951a7e0f7b8ffcff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-951a7e0f7b8ffcff"},"other_id":"n-b99042e296c8e644","other":{"id":"n-b99042e296c8e644","name":"one_plus_derivative of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-951a7e0f7b8ffcff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","5"],["39","80","25"]],"card_rows":[["1"],["8","5"],["39","80","25"],["162","585","600","125"]],"row_sums_preview":["1","13","144","1472","14336","135168","1245184","11272192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["D+1"],"edges":["e-0bf245e425ee01ec"],"text":"A013622 \u2192 D+1"},"headline":"A013622 \u2192 D+1","chain":null}},{"id":"l-3db73c835c0fa355","kind":"fingerprint_related","node_a_id":"n-2ae0099201780429","node_b_id":"n-3c7b6c13e5ec85c4","fingerprint":"f6-951a7e0f7b8ffcff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-951a7e0f7b8ffcff"},"other_id":"n-3c7b6c13e5ec85c4","other":{"id":"n-3c7b6c13e5ec85c4","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-951a7e0f7b8ffcff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["32","20"],["624","1280","400"]],"card_rows":[["1"],["32","20"],["624","1280","400"],["10368","37440","38400","8000"]],"row_sums_preview":["1","52","2304","94208","3670016","138412032","5100273664","184683593728"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013611","D+1"],"edges":["e-62a1b0a86f60f435","e-991a184d1289ea8c"],"text":"A038271 \u2192 basis A013611 \u2192 D+1"},"headline":"A038271 \u2192 basis A013611 \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:        156,          320,          100;\n  1:       1296,         4680,         4800,          1000;\n  2:       9936,        51840,        93600,         64000,          10000;\n  3:      72576,       496800,      1296000,       1560000,         800000,         100000;\n  4:     513216,      4354560,     14904000,      25920000,       23400000,        9600000,        1000000;\n  5:    3545856,     35925120,    152409600,     347760000,      453600000,      327600000,      112000000,       10000000;\n  6:   24074496,    283668480,   1437004800,    4064256000,     6955200000,     7257600000,     4368000000,     1280000000,      100000000;\n  7:  161243136,   2166704640,  12765081600,   43110144000,    91445760000,   125193600000,   108864000000,    56160000000,    14400000000,    1000000000;\n  8: 1068235776,  16124313600, 108335232000,  425502720000,  1077753600000,  1828915200000,  2086560000000,  1555200000000,   702000000000,  160000000000,   10000000000;\n  9: 7014076416, 117505935360, 886837248000, 3972291840000, 11701324800000, 23710579200000, 33530112000000, 32788800000000, 21384000000000, 8580000000000, 1760000000000, 100000000000;\n","recurrence_sentence":"The row polynomials satisfy (5*n+8+5*x)*P(n,x) = (30*n+78+(50*n+160)*x+50*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 156 + 320*x + 100*x^2.","mathematica":"P[0, x_] := 156 + 320*x + 100*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((30*n + 78 + (50*n + 160)*x + 50*x^2)*P[n - 1, x])/(5*n + 8 + 5*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 156 + 320*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((30*n + 78 + (50*n + 160)*x + 50*x^2)*v[i - 1])/(5*n + 8 + 5*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-bbfc5a54ff13a785","steps":["basis A013609","D+1"],"edges":["e-31086658f6c15e80","e-e1d4711e5a448783"],"text":"A013612 \u2192 basis A013609 \u2192 D+1"},"headline":"A013612 \u2192 basis A013609 \u2192 D+1","chain":null}
