{"id":"n-8ff9f44decd2285a","revision":0,"first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","name":"derivative of Triangle of coefficients in expansion of (1+6x)^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":[[],["6"],["12","72"],["18","216","648"],["24","432","2592","5184"],["30","720","6480","25920","38880"],["36","1080","12960","77760","233280","279936"],["42","1512","22680","181440","816480","1959552","1959552"],["48","2016","36288","362880","2177280","7838208","15676416","13436928"],["54","2592","54432","653184","4898880","23514624","70543872","120932352","90699264"],["60","3240","77760","1088640","9797760","58786560","235146240","604661760","906992640","604661760"],["66","3960","106920","1710720","17962560","129330432","646652160","2217093120","4988459520","6651279360","3990767616"],["72","4752","142560","2566080","30792960","258660864","1551965184","6651279360","19953838080","39907676160","47889211392","26121388032"],["78","5616","185328","3706560","50038560","480370176","3362591232","17293326336","64849973760","172933263360","311279874048","339578044416","169789022208"],["84","6552","235872","5189184","77837760","840647808","6725182464","40351094784","181579926528","605266421760","1452639412224","2377046310912","2377046310912","1097098297344"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","6","84","882","8232","72030","605052","4941258","39530064","311299254","2421216420","18643366434","142367525496","1079620401678","8138676874188"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["6"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["1","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182948,"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-b1a99f7ec702dba2"]},{"id":182949,"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-b1a99f7ec702dba2"]},{"id":182950,"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-b1a99f7ec702dba2"]},{"id":182951,"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-b1a99f7ec702dba2"]}],"region_exact":{},"incoming_edges":[{"id":"e-b1a99f7ec702dba2","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-8ff9f44decd2285a","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":"d5617bd1e993e73af52c65c351efcd8db57c8a7233bea002064e1da028ca676c","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-efce1b6c07b70e1a","role":"sequence","name":"Triangle of coefficients in expansion of (1+6x)^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-269dc9a9cf01dc43","kind":"fingerprint_related","node_a_id":"n-8ff9f44decd2285a","node_b_id":"n-64bc2cdf1204163e","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-64bc2cdf1204163e","other":{"id":"n-64bc2cdf1204163e","name":"shift_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","2"],["144","48","4"]],"card_rows":[["1"],["12","2"],["144","48","4"],["1728","864","144","8"]],"row_sums_preview":["1","14","196","2744","38416","537824","7529536","105413504"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","basis A007318"],"edges":["e-88ca34e9877f4f47","e-12984afc1d27ca59"],"text":"A038291 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A038291 \u2192 basis A013609 \u2192 basis A007318","chain":null}},{"id":"l-2823f4040f06de73","kind":"fingerprint_related","node_a_id":"n-8ff9f44decd2285a","node_b_id":"n-72192ea1b98c5050","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-72192ea1b98c5050","other":{"id":"n-72192ea1b98c5050","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["4","48","144"],["8","144","864","1728"],["16","384","3456","13824","20736"]],"card_rows":[["4","48","144"],["8","144","864","1728"],["16","384","3456","13824","20736"],["32","960","11520","69120","207360","248832"]],"row_sums_preview":["196","2744","38416","537824","7529536","105413504","1475789056","20661046784"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["reverse"],"edges":["e-896f56ac2a40747d"],"text":"A038328 \u2192 reverse"},"headline":"A038328 \u2192 reverse","chain":null}},{"id":"l-4c7e06e823dc1062","kind":"fingerprint_related","node_a_id":"n-8ff9f44decd2285a","node_b_id":"n-a927649f3968c00b","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-a927649f3968c00b","other":{"id":"n-a927649f3968c00b","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["48","8"],["864","288","24"],["13824","6912","1152","64"]],"card_rows":[["48","8"],["864","288","24"],["13824","6912","1152","64"],["207360","138240","34560","3840","160"]],"row_sums_preview":["56","1176","21952","384160","6453888","105413504","1686616064","26564203008"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["D"],"edges":["e-65f122626ee17875"],"text":"A038328 \u2192 D"},"headline":"A038328 \u2192 D","chain":null}},{"id":"l-7172792b51dace0a","kind":"fingerprint_related","node_a_id":"n-8ff9f44decd2285a","node_b_id":"n-95ff44822e25b07f","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-95ff44822e25b07f","other":{"id":"n-95ff44822e25b07f","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["324","108","9"],["5832","2916","486","27"],["104976","69984","17496","1944","81"]],"card_rows":[["324","108","9"],["5832","2916","486","27"],["104976","69984","17496","1944","81"],["1889568","1574640","524880","87480","7290","243"]],"row_sums_preview":["441","9261","194481","4084101","85766121","1801088541","37822859361","794280046581"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-5083e5230277d126","e-192fdbcaf1e01247"],"text":"A038329 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A038329 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-91476b9edbdaa8ad","kind":"fingerprint_related","node_a_id":"n-0817f7209fcd849d","node_b_id":"n-8ff9f44decd2285a","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-0817f7209fcd849d","other":{"id":"n-0817f7209fcd849d","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["12","2"]],"card_rows":[["1"],["12","2"],["108","36","3"],["864","432","72","4"]],"row_sums_preview":["0","1","14","147","1372","12005","100842","823543"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["D"],"edges":["e-e34388594b9483c9"],"text":"A038255 \u2192 D"},"headline":"A038255 \u2192 D","chain":null}},{"id":"l-9b719eae470dd390","kind":"fingerprint_related","node_a_id":"n-ea9b82d727693fb6","node_b_id":"n-8ff9f44decd2285a","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-ea9b82d727693fb6","other":{"id":"n-ea9b82d727693fb6","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","48","4"],["1728","864","144","8"],["20736","13824","3456","384","16"]],"card_rows":[["144","48","4"],["1728","864","144","8"],["20736","13824","3456","384","16"],["248832","207360","69120","11520","960","32"]],"row_sums_preview":["196","2744","38416","537824","7529536","105413504","1475789056","20661046784"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":[],"edges":[],"text":"A038328"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","chain":null}},{"id":"l-ad23b92a34af97da","kind":"fingerprint_related","node_a_id":"n-8ff9f44decd2285a","node_b_id":"n-155eef1a24bcc703","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-155eef1a24bcc703","other":{"id":"n-155eef1a24bcc703","name":"mobius_x_over_x_plus_one of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","12","36"],["1","18","108","216"],["1","24","216","864","1296"]],"card_rows":[["1","12","36"],["1","18","108","216"],["1","24","216","864","1296"],["1","30","360","2160","6480","7776"]],"row_sums_preview":["49","343","2401","16807","117649","823543","5764801","40353607"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["x \u21a6 x/(x+1)"],"edges":["e-43aab10e23af0f21"],"text":"A013612 \u2192 x \u21a6 x/(x+1)"},"headline":"A013612 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-c08c97673e152d19","kind":"fingerprint_related","node_a_id":"n-22693abd431a71f2","node_b_id":"n-8ff9f44decd2285a","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-22693abd431a71f2","other":{"id":"n-22693abd431a71f2","name":"Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","1"],["36","12","1"]],"card_rows":[["1"],["6","1"],["36","12","1"],["216","108","18","1"]],"row_sums_preview":["1","7","49","343","2401","16807","117649","823543"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038255"],"derivation":{"root":"n-22693abd431a71f2","steps":[],"edges":[],"text":"A038255"},"headline":"A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","chain":null}},{"id":"l-cf298d5d5e21703a","kind":"fingerprint_related","node_a_id":"n-8ff9f44decd2285a","node_b_id":"n-d19e6641dcda14df","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-d19e6641dcda14df","other":{"id":"n-d19e6641dcda14df","name":"shift_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","3"],["324","108","9"]],"card_rows":[["1"],["18","3"],["324","108","9"],["5832","2916","486","27"]],"row_sums_preview":["1","21","441","9261","194481","4084101","85766121","1801088541"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-011a72a323f5e83e","e-bcc80721f40fd9b6"],"text":"A147716 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A147716 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-d1ccc5be37854c0d","kind":"fingerprint_related","node_a_id":"n-efce1b6c07b70e1a","node_b_id":"n-8ff9f44decd2285a","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-efce1b6c07b70e1a","other":{"id":"n-efce1b6c07b70e1a","name":"Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","6"],["1","12","36"]],"card_rows":[["1"],["1","6"],["1","12","36"],["1","18","108","216"]],"row_sums_preview":["1","7","49","343","2401","16807","117649","823543"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A013613"],"derivation":{"root":"n-efce1b6c07b70e1a","steps":[],"edges":[],"text":"A013613"},"headline":"A013613: Triangle of coefficients in expansion of (1+6x)^n.","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:  6;\n  2: 12,   72;\n  3: 18,  216,   648;\n  4: 24,  432,  2592,   5184;\n  5: 30,  720,  6480,  25920,   38880;\n  6: 36, 1080, 12960,  77760,  233280,   279936;\n  7: 42, 1512, 22680, 181440,  816480,  1959552,  1959552;\n  8: 48, 2016, 36288, 362880, 2177280,  7838208, 15676416,  13436928;\n  9: 54, 2592, 54432, 653184, 4898880, 23514624, 70543872, 120932352, 90699264;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n+6*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 6*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] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((n + 6*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-efce1b6c07b70e1a","steps":["D"],"edges":["e-b1a99f7ec702dba2"],"text":"A013613 \u2192 D"},"headline":"A013613 \u2192 D","chain":null}