{"id":"n-7920008a31ba6c4f","revision":0,"first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","name":"one_plus_derivative of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= 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":[["3"],["27","18"],["135","324","108"],["567","2430","2916","648"],["2187","13608","29160","23328","3888"],["8019","65610","204120","291600","174960","23328"],["28431","288684","1180980","2449440","2624400","1259712","139968"],["98415","1194102","6062364","16533720","25719120","22044960","8817984","839808"],["334611","4723920","28658448","96997824","198404640","246903552","176359680","60466176","5038848"],["1121931","18068994","127545840","515852064","1309470624","2142770112","2222131968","1360488960","408146688","30233088"],["3720087","67315860","542069820","2550916800","7737780960","15713647488","21427701120","19046845440","10203667200","2720977920","181398528"],["12223143","245525742","2221423380","11925536040","42090127200","102138708672","172850122368","202032610560","157136474880","74826892800","17958454272","1088391168"],["39858075","880066296","8838926712","53314161120","214659648720","606097831680","1225664504064","1777886972928","1818293495040","1257091799040","538753628160","117546246144","6530347008"],["129140163","3108929850","34322585544","229812094512","1039626141840","3348690520032","7879271811840","13657404473856","17334397986048","15758543623680","9805316032512","3820252999680","764050599936","39182082048"],["416118303","10847773692","130575053700","961032395232","4826053984752","17465719182912","46881667280448","94551261742080","143402746975488","161787714536448","132371766438912","74876958793728","26741770997760","4936942338048","235092492288"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["3","45","567","6561","72171","767637","7971615","81310473","817887699","8135830269","80196041223","784526490225","7625597484987","73714109021541","709180566103791"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3"]],"recurrence":{"denominator":{"coeffs":[["-1","2"],["2","0"]]},"terms":[{"coeff":{"coeffs":[["3","12","12"],["6","12","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185776,"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-d0c057da145142a3"]},{"id":185777,"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-d0c057da145142a3"]},{"id":185778,"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-d0c057da145142a3"]},{"id":185779,"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-d0c057da145142a3"]}],"region_exact":{},"incoming_edges":[{"id":"e-d0c057da145142a3","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-7920008a31ba6c4f","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":"3f05bfcda6fa2b057feacff7eaf9dfaff8031662f5dff79f21a49342a4374336","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-219653841046440e","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= 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-272917c6f320e517","kind":"fingerprint_related","node_a_id":"n-7920008a31ba6c4f","node_b_id":"n-d468705beb32adb2","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-d468705beb32adb2","other":{"id":"n-d468705beb32adb2","name":"one_plus_derivative of basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["10","24","8"]],"card_rows":[["1"],["3","2"],["10","24","8"],["42","180","216","48"]],"row_sums_preview":["1","5","42","486","7128","126360","2624400","62460720"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013609","D+1"],"edges":["e-456dc355a6d78eff","e-d0271709f31646f7"],"text":"A136572 \u2192 basis A013609 \u2192 D+1"},"headline":"A136572 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-3b4946eb1dfc1611","kind":"fingerprint_related","node_a_id":"n-c84baea1c0b451a0","node_b_id":"n-7920008a31ba6c4f","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-c84baea1c0b451a0","other":{"id":"n-c84baea1c0b451a0","name":"one_plus_derivative of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["5","12","4"]],"card_rows":[["1"],["3","2"],["5","12","4"],["7","30","36","8"]],"row_sums_preview":["1","5","21","81","297","1053","3645","12393"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["D+1"],"edges":["e-d994c6a22dd93cd9"],"text":"A013609 \u2192 D+1"},"headline":"A013609 \u2192 D+1","chain":null}},{"id":"l-45ac6fc72aef9d19","kind":"fingerprint_related","node_a_id":"n-7e97a6745522f0af","node_b_id":"n-7920008a31ba6c4f","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-7e97a6745522f0af","other":{"id":"n-7e97a6745522f0af","name":"one_plus_derivative of basis_map_A013609 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","4"],["20","48","16"]],"card_rows":[["1"],["6","4"],["20","48","16"],["56","240","288","64"]],"row_sums_preview":["1","10","84","648","4752","33696","233280","1586304"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013609","D+1"],"edges":["e-6abeab86de871564","e-315c5cc768f7bb66"],"text":"A130123 \u2192 basis A013609 \u2192 D+1"},"headline":"A130123 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-49de224b2e100384","kind":"fingerprint_related","node_a_id":"n-7920008a31ba6c4f","node_b_id":"n-bf0bee765dd426d0","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-bf0bee765dd426d0","other":{"id":"n-bf0bee765dd426d0","name":"one_plus_derivative of basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","4"],["60","144","48"]],"card_rows":[["1"],["6","4"],["60","144","48"],["840","3600","4320","960"]],"row_sums_preview":["1","10","252","9720","498960","31842720","2424945600","214365191040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013611","D+1"],"edges":["e-20dcbad557ba463b","e-15c67c151e329f3e"],"text":"A085881 \u2192 basis A013611 \u2192 D+1"},"headline":"A085881 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-4c6f2bd5d80ac09b","kind":"fingerprint_related","node_a_id":"n-7920008a31ba6c4f","node_b_id":"n-de8788eb877ead40","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-de8788eb877ead40","other":{"id":"n-de8788eb877ead40","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["30","20"],["300","720","240"]],"card_rows":[["1"],["30","20"],["300","720","240"],["1960","8400","10080","2240"]],"row_sums_preview":["1","50","1260","22680","332640","4245696","48988800","523480320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013611","D+1"],"edges":["e-4faaf8fb320b6fca","e-20a87dc0c1d72e8a"],"text":"A178821 \u2192 basis A013611 \u2192 D+1"},"headline":"A178821 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-55df4548e693c8cc","kind":"fingerprint_related","node_a_id":"n-5560650c49185322","node_b_id":"n-7920008a31ba6c4f","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-5560650c49185322","other":{"id":"n-5560650c49185322","name":"one_plus_derivative of basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-3","-2"],["5","12","4"]],"card_rows":[["1"],["-3","-2"],["5","12","4"],["-7","-30","-36","-8"]],"row_sums_preview":["1","-5","21","-81","297","-1053","3645","-12393"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013609","D+1"],"edges":["e-66f8525b021adc45","e-57c147f3b71e7848"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013609 \u2192 D+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-629a8398b10ac2da","kind":"fingerprint_related","node_a_id":"n-1d41671801583ee9","node_b_id":"n-7920008a31ba6c4f","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-1d41671801583ee9","other":{"id":"n-1d41671801583ee9","name":"one_plus_derivative of basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-3","-2"],["5","12","4"]],"card_rows":[["1"],["-3","-2"],["5","12","4"],["-7","-30","-36","-8"]],"row_sums_preview":["1","-5","21","-81","297","-1053","3645","-12393"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013609","D+1"],"edges":["e-13dcb978bba71940","e-81832318dadb359f"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013609 \u2192 D+1"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-7064a7471fecf23e","kind":"fingerprint_related","node_a_id":"n-416caeeb4716bb0a","node_b_id":"n-7920008a31ba6c4f","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-416caeeb4716bb0a","other":{"id":"n-416caeeb4716bb0a","name":"one_plus_derivative of basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["5","12","4"]],"card_rows":[["1"],["3","2"],["5","12","4"],["7","30","36","8"]],"row_sums_preview":["1","5","21","81","297","1053","3645","12393"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013609","D+1"],"edges":["e-7869781294d5f3a8","e-cf2cf098c77be2aa"],"text":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 D+1"},"headline":"basis-map coefficients of A026729 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-7160649878f08678","kind":"fingerprint_related","node_a_id":"n-7920008a31ba6c4f","node_b_id":"n-650ba4a83be0a684","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-650ba4a83be0a684","other":{"id":"n-650ba4a83be0a684","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","8"],["80","192","64"]],"card_rows":[["1"],["12","8"],["80","192","64"],["448","1920","2304","512"]],"row_sums_preview":["1","20","336","5184","76032","1078272","14929920","203046912"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A013611","D+1"],"edges":["e-e6fcabb7b5383fc4","e-354c7d585dfc3d1e"],"text":"A038208 \u2192 basis A013611 \u2192 D+1"},"headline":"A038208 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-7221e28aa02194c2","kind":"fingerprint_related","node_a_id":"n-7920008a31ba6c4f","node_b_id":"n-82486c5ae756319a","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-82486c5ae756319a","other":{"id":"n-82486c5ae756319a","name":"basis_map_A013609 of Triangle read by rows, (n-1) zeros followed by (2n, 1).","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["5","12","4"]],"card_rows":[["1"],["3","2"],["5","12","4"],["7","30","36","8"]],"row_sums_preview":["1","5","21","81","297","1053","3645","12393"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-66d5ecabca128038","steps":["basis A013609"],"edges":["e-398458c0779c4b82"],"text":"A134082 \u2192 basis A013609"},"headline":"A134082 \u2192 basis A013609","chain":null}},{"id":"l-7b4c47036668a1dd","kind":"fingerprint_related","node_a_id":"n-2a85d054e06ab550","node_b_id":"n-7920008a31ba6c4f","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-2a85d054e06ab550","other":{"id":"n-2a85d054e06ab550","name":"one_plus_derivative of derivative of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["12","8"]],"card_rows":[["2"],["12","8"],["30","72","24"],["56","240","288","64"]],"row_sums_preview":["0","2","20","126","648","2970","12636","51030"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["D","D+1"],"edges":["e-9f3f778aa1812acb","e-1935b0828880ddb6"],"text":"A013609 \u2192 D \u2192 D+1"},"headline":"A013609 \u2192 D \u2192 D+1","chain":null}},{"id":"l-8b46c347b104070e","kind":"fingerprint_related","node_a_id":"n-7920008a31ba6c4f","node_b_id":"n-abb390907d328d80","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-abb390907d328d80","other":{"id":"n-abb390907d328d80","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","12"],["180","432","144"]],"card_rows":[["1"],["18","12"],["180","432","144"],["1512","6480","7776","1728"]],"row_sums_preview":["1","30","756","17496","384912","8188128","170061120","3469246848"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013611","D+1"],"edges":["e-fc7ebcb9132c171a","e-524c8289543c5fdc"],"text":"A038221 \u2192 basis A013611 \u2192 D+1"},"headline":"A038221 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-92390acfd97d5c15","kind":"fingerprint_related","node_a_id":"n-7920008a31ba6c4f","node_b_id":"n-e6c5544535083f78","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-e6c5544535083f78","other":{"id":"n-e6c5544535083f78","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["36","24"],["420","1008","336"]],"card_rows":[["1"],["36","24"],["420","1008","336"],["3136","13440","16128","3584"]],"row_sums_preview":["1","60","1764","36288","598752","8491392","107775360","1256352768"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013611","D+1"],"edges":["e-377b2cf400ddb478","e-cfc3c551e5c94ffb"],"text":"A178822 \u2192 basis A013611 \u2192 D+1"},"headline":"A178822 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-9f804c08107b74e8","kind":"fingerprint_related","node_a_id":"n-7920008a31ba6c4f","node_b_id":"n-986ee60520ca2004","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-986ee60520ca2004","other":{"id":"n-986ee60520ca2004","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","8"],["120","288","96"]],"card_rows":[["1"],["12","8"],["120","288","96"],["1120","4800","5760","1280"]],"row_sums_preview":["1","20","504","12960","332640","8491392","215550720","5444195328"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013611","D+1"],"edges":["e-6d76bdb8087cd7e0","e-9d1dc8bfa419b54b"],"text":"A356546 \u2192 basis A013611 \u2192 D+1"},"headline":"A356546 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-a8c7f83d3c77ed67","kind":"fingerprint_related","node_a_id":"n-c454e53dd7e5aa56","node_b_id":"n-7920008a31ba6c4f","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-c454e53dd7e5aa56","other":{"id":"n-c454e53dd7e5aa56","name":"one_plus_derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","6"],["15","36","12"]],"card_rows":[["3"],["9","6"],["15","36","12"],["21","90","108","24"]],"row_sums_preview":["3","15","63","243","891","3159","10935","37179"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["D+1"],"edges":["e-6290caa6786f877f"],"text":"A120909 \u2192 D+1"},"headline":"A120909 \u2192 D+1","chain":null}},{"id":"l-aab8966da984c025","kind":"fingerprint_related","node_a_id":"n-04bd7c60de15be94","node_b_id":"n-7920008a31ba6c4f","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-04bd7c60de15be94","other":{"id":"n-04bd7c60de15be94","name":"one_plus_derivative of basis_map_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["12","8"],["40","96","32"]],"card_rows":[["2"],["12","8"],["40","96","32"],["112","480","576","128"]],"row_sums_preview":["2","20","168","1296","9504","67392","466560","3172608"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013611","D+1"],"edges":["e-3deac81ab612cd2e","e-54d16e8dcca98281"],"text":"A028326 \u2192 basis A013611 \u2192 D+1"},"headline":"A028326 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-b4bd9524ac750525","kind":"fingerprint_related","node_a_id":"n-7920008a31ba6c4f","node_b_id":"n-9754b486dd6d7f50","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-9754b486dd6d7f50","other":{"id":"n-9754b486dd6d7f50","name":"one_plus_derivative of basis_map_A013611 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","8"],["120","288","96"]],"card_rows":[["1"],["12","8"],["120","288","96"],["1344","5760","6912","1536"]],"row_sums_preview":["1","20","504","15552","570240","24261120","1175731200","63959777280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013611","D+1"],"edges":["e-413ab29272bdfecc","e-2829512efac883b7"],"text":"A156992 \u2192 basis A013611 \u2192 D+1"},"headline":"A156992 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-bc2a4c3ae6a9cd76","kind":"fingerprint_related","node_a_id":"n-7920008a31ba6c4f","node_b_id":"n-d06c736e578e5bae","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-d06c736e578e5bae","other":{"id":"n-d06c736e578e5bae","name":"one_plus_derivative of basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","4"],["40","96","32"]],"card_rows":[["1"],["6","4"],["40","96","32"],["280","1200","1440","320"]],"row_sums_preview":["1","10","168","3240","66528","1415232","30792960","680524416"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013611","D+1"],"edges":["e-9f3d1c5f0f510549","e-9d73dd6f90e00a03"],"text":"A085880 \u2192 basis A013611 \u2192 D+1"},"headline":"A085880 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-c05778fa8a2d1d05","kind":"fingerprint_related","node_a_id":"n-21572076ef05df89","node_b_id":"n-7920008a31ba6c4f","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-21572076ef05df89","other":{"id":"n-21572076ef05df89","name":"one_plus_derivative of basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["5","12","4"],["7","30","36","8"],["9","56","120","96","16"]],"card_rows":[["5","12","4"],["7","30","36","8"],["9","56","120","96","16"],["11","90","280","400","240","32"]],"row_sums_preview":["21","81","297","1053","3645","12393","41553","137781"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013609","D+1"],"edges":["e-7bee75fc05394bb8","e-00e142eb1e265861"],"text":"basis-map coefficients of A112227 \u2192 basis A013609 \u2192 D+1"},"headline":"basis-map coefficients of A112227 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-d02e1bc22c6b2a43","kind":"fingerprint_related","node_a_id":"n-7920008a31ba6c4f","node_b_id":"n-8ea454af8ae1e8ed","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-8ea454af8ae1e8ed","other":{"id":"n-8ea454af8ae1e8ed","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["24","16"],["200","480","160"]],"card_rows":[["1"],["24","16"],["200","480","160"],["1120","4800","5760","1280"]],"row_sums_preview":["1","40","840","12960","166320","1886976","19595520","190356480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013611","D+1"],"edges":["e-159425c642d412bb","e-1fb1bad4660e02c3"],"text":"A178820 \u2192 basis A013611 \u2192 D+1"},"headline":"A178820 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-d245a64179201fda","kind":"fingerprint_related","node_a_id":"n-7f7ad0475cdb9085","node_b_id":"n-7920008a31ba6c4f","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-7f7ad0475cdb9085","other":{"id":"n-7f7ad0475cdb9085","name":"one_plus_derivative of basis_map_A013610 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","6"],["45","108","36"]],"card_rows":[["1"],["9","6"],["45","108","36"],["189","810","972","216"]],"row_sums_preview":["1","15","189","2187","24057","255879","2657205","27103491"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A013610","D+1"],"edges":["e-14a3a792ab5fe5af","e-b32dc623debdbafd"],"text":"A013609 \u2192 basis A013610 \u2192 D+1"},"headline":"A013609 \u2192 basis A013610 \u2192 D+1","chain":null}},{"id":"l-e834edbeada85deb","kind":"fingerprint_related","node_a_id":"n-7920008a31ba6c4f","node_b_id":"n-6e77991891ce5749","fingerprint":"f6-4be2d3b20c0d29e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4be2d3b20c0d29e0"},"other_id":"n-6e77991891ce5749","other":{"id":"n-6e77991891ce5749","name":"one_plus_derivative of basis_map_A013611 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-4be2d3b20c0d29e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","4"],["40","96","32"]],"card_rows":[["1"],["6","4"],["40","96","32"],["336","1440","1728","384"]],"row_sums_preview":["1","10","168","3888","114048","4043520","167961600","7994972160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013611","D+1"],"edges":["e-b8ab5069d7db191f","e-84d8e90e868bed92"],"text":"A196347 \u2192 basis A013611 \u2192 D+1"},"headline":"A196347 \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:       3;\n  1:      27,       18;\n  2:     135,      324,       108;\n  3:     567,     2430,      2916,       648;\n  4:    2187,    13608,     29160,     23328,       3888;\n  5:    8019,    65610,    204120,    291600,     174960,      23328;\n  6:   28431,   288684,   1180980,   2449440,    2624400,    1259712,     139968;\n  7:   98415,  1194102,   6062364,  16533720,   25719120,   22044960,    8817984,     839808;\n  8:  334611,  4723920,  28658448,  96997824,  198404640,  246903552,  176359680,   60466176,   5038848;\n  9: 1121931, 18068994, 127545840, 515852064, 1309470624, 2142770112, 2222131968, 1360488960, 408146688, 30233088;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1+2*x)*P(n,x) = (6*n+3+(12*n+12)*x+12*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3.","mathematica":"P[0, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 3 + (12*n + 12)*x + 12*x^2)*P[n - 1, x])/(2*n - 1 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 3 + (12*n + 12)*x + 12*x^2)*v[i - 1])/(2*n - 1 + 2*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-765f9e2b013b3c68","steps":["basis A013610","D+1"],"edges":["e-284e6c48cfe1aa93","e-d0c057da145142a3"],"text":"A120909 \u2192 basis A013610 \u2192 D+1"},"headline":"A120909 \u2192 basis A013610 \u2192 D+1","chain":null}